ACT Exponential and Logarithmic Functions: Growth, Decay, Logs, and Inverse Relationships

The ACT current Math description explicitly includes:

• exponential relationships;
• logarithmic functions;
• function manipulation and graph interpretation.

The central idea is that exponential functions and logarithms are inverses.

$$\boxed{b^x=y\iff\log_b(y)=x}$$

That one relationship connects most of the topic.

Exponential functions

Basic form:

$$y=ab^x$$

where:

• $a$ is initial value;
• $b$ is growth/decay factor.

If:

$$b>1$$

growth.

If:

$$0<b<1$$

decay.

Growth factor vs growth rate

If a quantity grows by 8% per period:

$$r=0.08$$

growth factor:

$$1+r=1.08$$

Model:

$$\boxed{A=A_0(1.08)^t}$$

Do not use:

$$0.08^t$$

That would shrink the quantity drastically.

Decay factor

If a quantity decreases by 12% per period:

$$1-0.12=0.88$$

Model:

$$\boxed{A=A_0(0.88)^t}$$

The factor is what remains after each period.

Worked growth example

Population:

$$P_0=500$$

grows 6% annually.

After 3 years:

$$P=500(1.06)^3$$ $$P\approx595.5$$

Depending on context:

about 596.

Repeated percent change is multiplicative

Two 10% increases do not mean 20% of the original added once unless you are approximating.

Exact:

$$1.10^2=1.21$$

So total increase:

$$21\%$$

Repeated change compounds.

Exponential vs linear

Linear:

$$y=a+dx$$

adds constant amount.

Exponential:

$$y=ab^x$$

multiplies by constant factor.

Table:

Differences:

5, 10, 20

not constant.

Ratios:

2, 2, 2

constant.

So exponential.

Exponential equations with same base

$$3^{x+1}=3^5$$

Same base:

$$x+1=5$$ $$\boxed{x=4}$$

Always look for a common-base rewrite before using logarithms.

Rewrite bases

$$8^x=4^{x+1}$$ $$(2^3)^x=(2^2)^{x+1}$$ $$2^{3x}=2^{2x+2}$$ $$x=2$$

Efficient ACT method.

What is a logarithm?

$$\log_b(y)=x$$

means:

$$b^x=y$$

Example:

$$\log_2(8)=3$$

because:

$$2^3=8$$

Logarithms answer:

what exponent produces this number?

Common logarithm

If no base is written:

$$\log(100)$$

usually means base 10:

$$\log_{10}(100)=2$$

because:

$$10^2=100$$

Natural logarithm

$$\ln(x)$$

means logarithm base:

$$e$$

You may not need advanced $e$-based modeling, but you should recognize the notation.

Evaluate logs by rewriting exponentially

$$\log_3(81)=?$$

Ask:

$$3^{?}=81$$ $$3^4=81$$

So:

$$\boxed{4}$$

Log of 1

For any valid base:

$$\log_b(1)=0$$

because:

$$b^0=1$$

Log of the base

$$\log_b(b)=1$$

because:

$$b^1=b$$

These are fast points.

Product rule

$$\log_b(MN) = \log_b(M)+\log_b(N)$$

Quotient rule

$$\log_b\left(\frac{M}{N}\right) = \log_b(M)-\log_b(N)$$

Power rule

$$\log_b(M^k)=k\log_b(M)$$

These are algebraic identities, not arbitrary memorization.

The dangerous false rule

Wrong:

$$\log(a+b)=\log a+\log b$$

There is no general log-of-sum rule.

This is similar to the radical trap:

$$\sqrt{a+b}\ne\sqrt a+\sqrt b$$

Solve a simple log equation

$$\log_2(x)=5$$

Rewrite:

$$x=2^5$$ $$\boxed{x=32}$$

Solve with log rules

$$\log_3(x)+\log_3(9)=4$$

Since:

$$\log_3(9)=2$$

then:

$$\log_3(x)=2$$ $$x=9$$

Or combine product:

$$\log_3(9x)=4$$ $$9x=81$$ $$x=9$$

Domain of logarithm

For real-valued logarithms:

$$\boxed{\text{argument}>0}$$

So:

$$\log(x-3)$$

requires:

$$x>3$$

Not $x\ge3$.

At:

$$x=3$$

argument is zero, and:

$$\log(0)$$

is undefined.

Domain trap in equations

Suppose algebra produces:

$$x=2$$

for:

$$\log(x-5)=...$$

Reject it because:

$$x-5<0$$

Domain comes first.

Exponential graph features

For:

$$y=b^x,\quad b>1$$

key features:

• y-intercept $(0,1)$;
• domain: all real x;
• range: $y>0$;
• horizontal asymptote: $y=0$;
• increasing.

If multiplied/shifted, these features move.

Logarithmic graph features

For:

$$y=\log_b(x),\quad b>1$$

key features:

• x-intercept $(1,0)$;
• domain: $x>0$;
• range: all real y;
• vertical asymptote: $x=0$;
• increasing.

Exponential and log graphs are reflections across:

$$y=x$$

because they are inverses.

Transformations

$$y=2^{x-3}+4$$

from:

$$y=2^x$$

shift:

• right 3;
• up 4.

Horizontal asymptote becomes:

$$y=4$$

For:

$$y=\log(x+2)-1$$

shift:

• left 2;
• down 1.

Vertical asymptote becomes:

$$x=-2$$

See ACT Function Composition and Transformations.

Doubling time

If growth follows:

$$A=A_0(2)^{t/d}$$

then $d$ is doubling time.

Example:

doubles every 5 years.

$$A=A_0 2^{t/5}$$

After 15 years:

$$A=A_0 2^3=8A_0$$

Half-life

If quantity halves every $h$ units:

$$A=A_0\left(\frac12\right)^{t/h}$$

Example:

half-life = 4 hours.

After 12 hours:

$$A=A_0\left(\frac12\right)^3 = \frac{A_0}{8}$$

Percent change from factor

If model uses:

$$1.07^t$$

growth rate:

7%.

If model uses:

$$0.82^t$$

decay rate:

$$1-0.82=0.18$$

or:

18%.

Do not confuse factor with percent.

Sequence connection

Geometric sequence:

$$a_n=a_1r^{n-1}$$

is exponential in index $n$.

Common ratio:

$r$.

The ACT can move between sequences and function models.

Worked example: identify model

Table:

Ratio:

$$\frac{180}{200}=0.9$$

constant.

Model:

$$\boxed{200(0.9)^t}$$

Decay:

10% per period.

Worked example: solve exponent by rewriting

$$9^{x}=27$$

Write base 3:

$$3^{2x}=3^3$$ $$2x=3$$ $$\boxed{x=\frac32}$$

No logs needed.

Worked example: use logs conceptually

If:

$$5^x=17$$

no simple common base.

Then:

$$x=\log_5(17)$$

or:

$$x=\frac{\ln17}{\ln5}$$

if change-of-base is needed.

ACT may provide a calculator context where numerical approximation is appropriate.

Change-of-base formula

$$\boxed{\log_b(a)=\frac{\log(a)}{\log(b)}}$$

or:

$$\frac{\ln(a)}{\ln(b)}$$

Useful when calculator has only $\log$ and $\ln$.

Common ACT traps

Trap 1

Using percent rate instead of growth factor.

Trap 2

Calling a constant difference exponential.

Trap 3

Calling a constant ratio linear.

Trap 4

Forgetting log domain.

Trap 5

Using fake log sum rule.

Trap 6

Using logs when common-base rewrite is faster.

Trap 7

Confusing exponential asymptote $y=0$ with log asymptote $x=0$.

Fast ACT routine

Growth/decay

Find initial value and factor.

Exponential equation

Try common base first.

Log evaluation

Rewrite exponentially.

Log equation

Check domain.

Graph

Identify intercept, asymptote, shift.

Current ACT relevance

Functions are about 17–20% of current ACT Math, and ACT explicitly includes logarithmic functions as well as linear, radical, piecewise, and polynomial functions. Algebra also explicitly includes exponential relationships.

Practice set

1

5% annual growth:

$$\boxed{A=A_0(1.05)^t}$$

2

20% annual decay:

$$\boxed{A=A_0(0.8)^t}$$

3

$$\log_5(125)=\boxed{3}$$

4

$$\log_2(x)=6$$ $$\boxed{x=64}$$

How to review mistakes

Bad note:

exponential question wrong.

Better:

I used 0.12 as the growth factor instead of 1.12.

Or:

I used logarithms even though both sides could be rewritten as powers of 2.

Exponential growth from two data points

Suppose an exponential function has:

$$f(0)=200$$

and:

$$f(3)=675$$

Assume:

$$f(t)=200b^t$$

Then:

$$675=200b^3$$ $$b^3=3.375$$ $$b=1.5$$

So:

$$\boxed{f(t)=200(1.5)^t}$$

The factor can be inferred from data, not just stated in a word problem.

Exponential decay and percent remaining

Suppose 70% remains each period.

Factor:

$$0.70$$

Decay rate:

$$1-0.70=0.30$$

or:

30%.

Students often confuse:

percent remaining

with:

percent lost.

Read the wording carefully.

Continuous vs discrete exponential models

A model such as:

$$A=A_0b^t$$

works naturally for repeated periods.

If the question gives:

• yearly growth;
• monthly decay;
• doubling every fixed interval,

this form is usually enough.

Do not introduce continuous-growth formulas unless the problem actually requires them.

Logarithms undo exponentials

If:

$$y=3^x$$

then inverse:

$$x=\log_3(y)$$

Graphically:

the exponential and log graphs reflect across $y=x$.

This explains why:

• exponential domain is all real;
• exponential range is positive;
• log domain is positive;
• log range is all real.

The domain/range swap because inverses swap x and y.

Solve an exponential equation with logs

$$2^x=15$$

Take log:

$$x\log2=\log15$$ $$x=\frac{\log15}{\log2}$$

This is the change-of-base idea appearing inside equation solving.

Logarithmic equations with multiple terms

$$\log(x)+\log(x-9)=1$$

Assume common log.

Combine:

$$\log[x(x-9)]=1$$

Rewrite:

$$x(x-9)=10$$ $$x^2-9x-10=0$$ $$(x-10)(x+1)=0$$

Candidates:

$$10,-1$$

Domain requires:

$$x>9$$

So:

$$\boxed{x=10}$$

The domain check removes the extraneous algebraic candidate.

Log subtraction example

$$\log_2(x)-\log_2(x-2)=2$$

Combine:

$$\log_2\left(\frac{x}{x-2}\right)=2$$

Rewrite:

$$\frac{x}{x-2}=4$$ $$x=4x-8$$ $$x=\frac83$$

Domain:

$$x>2$$

Valid.

Graph comparison: exponential vs logarithm

If a graph:

• crosses y-axis;
• approaches horizontal asymptote;
• grows rapidly,

likely exponential.

If it:

• has vertical asymptote;
• crosses x-axis;
• grows slowly,

likely logarithmic.

This qualitative recognition can be faster than reconstructing equations.

Transformations change asymptotes

For:

$$y=3^x-5$$

horizontal asymptote:

$$y=-5$$

For:

$$y=\log(x-4)+2$$

vertical asymptote:

$$x=4$$

Asymptotes often reveal the shift immediately.

Compare exponential models

Model A:

$$100(1.04)^t$$

Model B:

$$80(1.08)^t$$

A starts higher.

B grows faster.

Do not assume the larger growth rate means B is larger at every time.

ACT can ask:

when do the models cross?

That becomes an equation.

Compound-interest structure

A common model:

$$A=P\left(1+\frac rn\right)^{nt}$$

where:

• $P$: principal;
• $r$: annual rate;
• $n$: compounds/year;
• $t$: years.

You may be given the formula. The skill is interpreting variables and using it correctly.

Review checklist

When an exponential/log problem is wrong, ask:

• rate vs factor?
• constant difference vs ratio?
• common-base opportunity missed?
• wrong log inverse conversion?
• log-domain violation?
• wrong asymptote shift?
• base vs exponent confused?

These categories are far more actionable than "log mistake."

Solve for time in an exponential model

Suppose:

$$A=500(1.08)^t$$

and the question asks when the amount reaches 800.

Set:

$$800=500(1.08)^t$$ $$1.6=(1.08)^t$$

If a simple power is not obvious, use logarithms:

$$t=\frac{\log(1.6)}{\log(1.08)}$$

The important setup is:

$$\boxed{\text{target}=\text{initial}\times(\text{factor})^t}$$

Compare linear and exponential models over time

Linear:

$$L(t)=500+40t$$

Exponential:

$$E(t)=500(1.06)^t$$

Both start at 500.

At first, the linear model may be larger.

Eventually, exponential growth can overtake it because the increase itself grows.

ACT may ask which model better matches a table, graph, or long-term description.

Logarithmic scale interpretation

A logarithmic relationship compresses large multiplicative changes.

Even if ACT does not ask about a named real-world log scale, you should understand:

$$\log_b(b^k)=k$$

So equal steps in log output correspond to multiplicative steps in the original quantity.

That is the core inverse idea.

Domain and asymptote together

For:

$$y=\log(x+5)$$

domain:

$$x>-5$$

vertical asymptote:

$$x=-5$$

These are two expressions of the same boundary.

For:

$$y=2^x-7$$

range:

$$y>-7$$

horizontal asymptote:

$$y=-7$$

Tracking the boundary often reveals the graph immediately.

FAQ: ACT exponentials and logarithms

Are logarithms on the ACT?

Yes. Current ACT Functions content explicitly includes logarithmic functions.

What is the easiest way to understand a logarithm?

$$\log_b(y)=x$$

means:

$$b^x=y$$

How do I identify exponential data?

Look for a constant ratio rather than a constant difference.

What is the domain of a log function?

Its argument must be positive.

Should I always use logs to solve exponential equations?

No. Rewrite to a common base first when possible.

Related MastaPrep guides

• ACT Functions
• ACT Exponents and Radicals
• ACT Function Composition and Transformations
• ACT Algebra
• Enhanced ACT 2026–27 Guide

Final takeaway

The key inverse relationship is:

$$\boxed{b^x=y\iff\log_b(y)=x}$$

For modeling:

$$\boxed{\text{initial value}\times(\text{growth/decay factor})^t}$$

For ACT efficiency:

• look for constant ratios;
• convert percent to factor correctly;
• rewrite bases before using logs;
• check log domains;
• track asymptotes and shifts.

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Official ACT references:
https://www.act.org/content/act/en/products-and-services/the-act/test-preparation/description-of-math-test.html
https://www.act.org/content/dam/act/unsecured/documents/R2519-Design-Framework-for-the-ACT-Enhancements-2026-02.pdf

Trademark note: ACT® is a registered trademark of ACT Education Corp. MastaPrep is independent and is not affiliated with or endorsed by ACT Education Corp.