Solving Exponential and Logarithmic Equations

Q: How can I study Solving Exponential and Logarithmic Equations effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Q: Is this Solving Exponential and Logarithmic Equations study guide free?

Yes — all study notes, flashcards, and practice problems for Solving Exponential and Logarithmic Equations on Study Mondo are free to access. No account is needed.

Q: What course covers Solving Exponential and Logarithmic Equations?

Solving Exponential and Logarithmic Equations is part of the AP Precalculus course on Study Mondo, specifically in the Exponential and Logarithmic Functions section. You can explore the full course for more related topics and practice resources.

Q: Are there practice problems for Solving Exponential and Logarithmic Equations?

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Techniques for solving equations involving exponentials and logarithms

Written and reviewed by the Study Mondo Education TeamLast updated April 18, 2026

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Solving Exponential and Logarithmic Equations

Solving Exponential Equations

An exponential equation has the variable in the exponent.

Strategy 1: Same Base Method

If you can express both sides with the same base, set the exponents equal.

If $b^x = b^y$ , then $x = y$

Example

$2^{3x} = 2^{15}$ $3x = 15$ $x = 5$

Strategy 2: Taking Logarithms

If you can't easily get the same base, take the logarithm of both sides.

Steps:

Isolate the exponential expression
Take log (or ln) of both sides
Use the power rule: $\log(b^x) = x\log(b)$
Solve for the variable

Example

$5^x = 17$ $\ln(5^x) = \ln(17)$ $x\ln(5) = \ln(17)$ $x = \frac{\ln(17)}{\ln(5)}$

Solving Logarithmic Equations

A logarithmic equation contains logarithmic expressions.

Strategy 1: Convert to Exponential Form

Use the definition: $\log_b(x) = y$ means $b^y = x$

Example

$\log_3(x) = 4$ $x = 3^4 = 81$

Strategy 2: Combine Logarithms

Use logarithm properties to combine into a single log, then solve.

Key Properties to Use

$\log_b(M) + \log_b(N) = \log_b(MN)$
$\log_b(M) - \log_b(N) = \log_b(M/N)$
$p\log_b(M) = \log_b(M^p)$

Strategy 3: Equal Logs Method

If $\log_b(M) = \log_b(N)$ , then $M = N$

(assuming same base and same domain)

Important Reminders

⚠️ Check your answers!

Logarithms require positive arguments: $x > 0$ for $\log(x)$
Reject any solutions that give $\log(\text{negative})$ or $\log(0)$

⚠️ One-to-one property

$b^x$ is one-to-one (equal outputs → equal inputs)
$\log_b(x)$ is one-to-one (equal outputs → equal inputs)

Common Equations to Recognize

Type 1: $b^x = a$

$x = \log_b(a) = \frac{\ln(a)}{\ln(b)}$

Type 2: $a \cdot b^{cx} + d = e$

Isolate: $b^{cx} = \frac{e - d}{a}$
Take log: $cx\ln(b) = \ln\left(\frac{e - d}{a}\right)$
Solve: $x = \frac{1}{c\ln(b)}\ln\left(\frac{e - d}{a}\right)$

Type 3: $\log(x) + \log(x - 3) = 1$

Combine: $\log(x(x - 3)) = 1$
Convert: $x(x - 3) = 10^1$
Solve quadratic: $x^2 - 3x - 10 = 0$

Applications

Compound Interest: $A = P(1 + r)^t$
Exponential Growth/Decay: $A = A_0e^{kt}$
Half-life Problems: $A = A_0(1/2)^{t/h}$
Doubling Time: Solve $2A_0 = A_0e^{kt}$ for $t$

📚 Practice Problems

1Problem 1easy

❓ Question:

Solve for $x$ : $3^{2x-1} = 27$

💡 Show Solution

Solution:

Step 1: Express both sides with the same base.

Notice that $27 = 3^3$ : $3^{2x-1} = 3^3$

Step 2: Set exponents equal.

Since the bases are equal: $2x - 1 = 3$

Step 3: Solve for $x$ . $2x = 4$ $x = 2$

Step 4: Check. $3^{2(2)-1} = 3^{4-1} = 3^3 = 27$ ✓

Answer: $x = 2$

2Problem 2medium

❓ Question:

Solve the following equations:

a) $5^{2x-1} = 125$ b) $3e^{4x} = 24$ c) $\log_2(x+3) + \log_2(x-3) = 4$

💡 Show Solution

Solution:

Part (a): $5^{2x-1} = 125$

Rewrite 125 as a power of 5: $125 = 5^3$

$5^{2x-1} = 5^3$

Since the bases are equal: $2x - 1 = 3$

$2x = 4$

$x = 2$

Part (b): $3e^{4x} = 24$

$e^{4x} = 8$

Take natural log of both sides: $\ln(e^{4x}) = \ln 8$

$4x = \ln 8$

$x = \frac{\ln 8}{4} = \frac{2.079}{4} \approx 0.520$

Part (c): $\log_2(x+3) + \log_2(x-3) = 4$

Use product rule: $\log_2[(x+3)(x-3)] = 4$

$\log_2(x^2 - 9) = 4$

Convert to exponential form: $x^2 - 9 = 2^4 = 16$

$x^2 = 25$

$x = \pm 5$

Check domain: We need $x + 3 > 0$ and $x - 3 > 0$ , so $x > 3$ .

Therefore: $x = 5$ (reject $x = -5$ )

3Problem 3medium

❓ Question:

Solve for $x$ : $5^x = 23$

💡 Show Solution

Solution:

Step 1: Take the natural log of both sides.

$\ln(5^x) = \ln(23)$

Step 2: Use the power rule. $x\ln(5) = \ln(23)$

Step 3: Solve for $x$ . $x = \frac{\ln(23)}{\ln(5)}$

Step 4: Calculate (optional). $x \approx \frac{3.135}{1.609} \approx 1.948$

Answer: $x = \frac{\ln(23)}{\ln(5)} \approx 1.948$

4Problem 4medium

❓ Question:

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

💡 Show Solution

Solution:

Step 1: Combine logarithms using the product rule.

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

Step 2: Convert to exponential form. $x(x - 3) = 2^2 = 4$

Step 3: Expand and rearrange. $x^2 - 3x = 4$ $x^2 - 3x - 4 = 0$

Step 4: Factor. $(x - 4)(x + 1) = 0$ $x = 4 \text{ or } x = -1$

Step 5: Check both solutions in the original equation.

For $x = 4$ : $\log_2(4) + \log_2(4 - 3) = \log_2(4) + \log_2(1)$ $= 2 + 0 = 2$ ✓

For $x = -1$ : $\log_2(-1) + \log_2(-4)$ This is undefined (cannot take log of negative numbers) ✗

Answer: $x = 4$ (reject $x = -1$ )

5Problem 5hard

❓ Question:

Solve: log₂(x + 3) + log₂(x - 3) = 4

💡 Show Solution

Step 1: Use product rule to combine logs: log₂[(x + 3)(x - 3)] = 4

Step 2: Simplify: log₂(x² - 9) = 4

Step 3: Convert to exponential form: x² - 9 = 2⁴ x² - 9 = 16

Step 4: Solve for x: x² = 25 x = ±5

Step 5: Check domain restrictions: For x = 5: x + 3 = 8 > 0 ✓, x - 3 = 2 > 0 ✓ For x = -5: x + 3 = -2 < 0 ✗ x = -5 is extraneous

Step 6: Verify x = 5: log₂(8) + log₂(2) = 3 + 1 = 4 ✓

Answer: x = 5

Explain using:

⚠️ Common Mistakes: Solving Exponential and Logarithmic Equations

Avoid these 4 frequent errors

🌍 Real-World Applications: Solving Exponential and Logarithmic Equations

See how this math is used in the real world

📝 Worked Example: Related Rates — Expanding Circle

Problem:

A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of $2$ cm/s. How fast is the area of the circle increasing when the radius is $10$ cm?

2Write the relationship between variables

3Differentiate both sides with respect to time

4Substitute known values

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Logarithmic and Exponential Models

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📌 Related Topics in Exponential and Logarithmic Functions

Exponential Functions and Modeling Logarithmic Functions and Equations Exponential Functions and Growth/Decay Logarithmic Functions Logarithmic and Exponential Models

❓ Frequently Asked Questions

What is Solving Exponential and Logarithmic Equations?▾

Techniques for solving equations involving exponentials and logarithms

How can I study Solving Exponential and Logarithmic Equations effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Solving Exponential and Logarithmic Equations study guide free?▾

Yes — all study notes, flashcards, and practice problems for Solving Exponential and Logarithmic Equations on Study Mondo are free to access. No account is needed.

What course covers Solving Exponential and Logarithmic Equations?▾

Solving Exponential and Logarithmic Equations is part of the AP Precalculus course on Study Mondo, specifically in the Exponential and Logarithmic Functions section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Solving Exponential and Logarithmic Equations?▾

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Solving Exponential and Logarithmic Equations

Try the Interactive Version!

Solving Exponential and Logarithmic Equations

Solving Exponential Equations

Strategy 1: Same Base Method

Example

Strategy 2: Taking Logarithms

Example

Solving Logarithmic Equations

Strategy 1: Convert to Exponential Form

Example

Strategy 2: Combine Logarithms

Key Properties to Use

Strategy 3: Equal Logs Method

Important Reminders

Common Equations to Recognize

Type 1: bx=ab^x = abx=a

Type 2: a⋅bcx+d=ea \cdot b^{cx} + d = ea⋅bcx+d=e

Type 3: log⁡(x)+log⁡(x−3)=1\log(x) + \log(x - 3) = 1log(x)+log(x−3)=1

Applications

📚 Practice Problems

1Problem 1easy

❓ Question:

2Problem 2medium

❓ Question:

3Problem 3medium

❓ Question:

4Problem 4medium

❓ Question:

5Problem 5hard

❓ Question:

⚠️ Common Mistakes: Solving Exponential and Logarithmic Equations

🌍 Real-World Applications: Solving Exponential and Logarithmic Equations

📝 Worked Example: Related Rates — Expanding Circle

Practice Lab

Practice with Flashcards

Browse All Topics

📌 Related Topics in Exponential and Logarithmic Functions

❓ Frequently Asked Questions

Type 1: $b^x = a$

Type 2: $a \cdot b^{cx} + d = e$

Type 3: $\log(x) + \log(x - 3) = 1$