New YorkMathsSyllabus dot point

How do you solve exponential and logarithmic equations, and how do you model growth and decay?

Solve exponential equations (matching bases or taking logarithms) and logarithmic equations (condensing then rewriting in exponential form), check for extraneous solutions, and model exponential growth, decay, and compound interest.

A NY Regents Algebra II answer on solving exponential and logarithmic equations: matching bases, taking logs, condensing and rewriting logs, extraneous solutions, and modeling growth, decay, and compound interest.

Generated by Claude Opus 4.810 min answerUpdated 2026-06-02

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What this topic is asking
Exponential equations: match bases first
Exponential equations: take a log
Logarithmic equations: condense then rewrite
Modeling growth, decay, and interest
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What this topic is asking

The Regents Algebra II exam (the Creating Equations, A-CED, and Linear, Quadratic, and Exponential Models, F-LE, clusters) wants you to solve exponential equations (by matching bases or taking logarithms) and logarithmic equations (by condensing then rewriting in exponential form), to check for extraneous solutions, and to model growth, decay, and compound interest. This is one of the most reliable constructed-response topics on the exam.

Exponential equations: match bases first

The cleanest method, when it works, is to write both sides with a common base and set the exponents equal.

8^{x} = 32 \;\Rightarrow\; (2^3)^x = 2^5 \;\Rightarrow\; 2^{3x} = 2^5 \;\Rightarrow\; 3x = 5 \;\Rightarrow\; x = \tfrac{5}{3}.

This gives an exact answer with no logarithm. Always check whether both numbers are powers of a common base before reaching for logs.

Exponential equations: take a log

When the bases cannot be matched, take a logarithm of both sides and use the power property $\log(b^k) = k\log b$ to free the exponent.

5^{x} = 40 \;\Rightarrow\; \log 5^x = \log 40 \;\Rightarrow\; x\log 5 = \log 40 \;\Rightarrow\; x = \frac{\log 40}{\log 5}.

Either $\log$ or $\ln$ works. The result is exact in logarithmic form, and you can evaluate it on a calculator. The most common error is writing $\frac{\log 40}{\log 5}$ as $\log\frac{40}{5} = \log 8$ , which is wrong; the quotient of two logs is not the log of a quotient.

Logarithmic equations: condense then rewrite

For a logarithmic equation, condense to a single log using the properties, then rewrite in exponential form to remove the logarithm.

Solving a logarithmic equation with a check

Solve $\log_2(x) + \log_2(x - 2) = 3$ .

step 1 Condense the left side

Use the product property: $\log_2\big(x(x - 2)\big) = 3$ .

step 2 Rewrite in exponential form

$\log_2(\text{argument}) = 3$ means argument $= 2^3 = 8$ , so $x(x - 2) = 8$ .

step 3 Solve the quadratic

$x^2 - 2x - 8 = 0$ , which factors to $(x - 4)(x + 2) = 0$ , giving candidates $x = 4$ and $x = -2$ .

step 4 Check the domain

A log needs a positive argument. For $x = 4$ : $\log_2 4$ and $\log_2 2$ are both defined, valid. For $x = -2$ : $\log_2(-2)$ is undefined, so $x = -2$ is extraneous. The only solution is $x = 4$ .

Modeling growth, decay, and interest

Exponential models recur in word problems. Growth and decay use $A = a(1 \pm r)^t$ , and compound interest has its own form.

A clarifying point that protects credit is that finding when a quantity reaches a target requires logarithms: set the model equal to the target, isolate the exponential, and take a log to solve for the time $t$ . The Regents commonly pairs an evaluation part (substitute a time, compute a value) with a logarithmic part (solve for the time), and the second part is where the log technique earns the credits. Keep the logs exact until the final rounding step.

Try this

Q1. Solve $2^x = 64$ by matching bases. [1 credit]

Cue. $64 = 2^6$ , so $x = 6$ .

Q2. Solve $\log_3(x) = 4$ . [1 credit]

Cue. Exponential form: $x = 3^4 = 81$ .

Exam-style practice questions

Practice questions written in the style of NYSED exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Regents (style)2 marksPart I (multiple choice). What is the solution to

3^{2x} = 27

? (1)

x = 1.5

(2)

x = 3

(3)

x = 9

(4)

x = 13.5

Show worked answer →

The correct answer is (1).

Match bases: $27 = 3^3$ , so $3^{2x} = 3^3$ , which gives $2x = 3$ and $x = 1.5$ . When both sides can be written with the same base, set the exponents equal, with no logarithm needed. Choice (2) forgets to halve after setting $2x = 3$ .

Regents (style)4 marksPart III (constructed response). An investment of

2{,}000 earns 4% annual interest compounded annually, modeled by

A = 2000(1.04)^t

. (a) Find the value after 5 years to the nearest cent. (b) Use logarithms to find, to the nearest tenth of a year, when the investment doubles to

4{,}000.

Show worked answer →

A 4-credit question: credit for the evaluation and for the logarithmic solving.

(a) $A = 2000(1.04)^5 \approx 2000(1.2167) \approx \$ 2433.31$.
(b) Set $4000 = 2000(1.04)^t$ , so $2 = (1.04)^t$ . Take logs: $\log 2 = t\log 1.04$ , giving $t = \frac{\log 2}{\log 1.04} \approx \frac{0.3010}{0.0170} \approx 17.7$ years. Dividing $\log 2$ by $\log 1.04$ incorrectly (for example computing $\log(2/1.04)$ ) or rounding the logs too early loses credit.

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Sources & how we know this

Educator Guide to the Regents Examination in Algebra II — NYSED (2025)
New York State Next Generation Mathematics Learning Standards — NYSED (2017)