Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Solving Exponential and Logarithmic Equations

College Algebra

6. Exponential & Logarithmic Functions

Solving Exponential and Logarithmic Equations

Previous problemNext problem

1:42 minutes

Problem 59

Textbook Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log4(3x+2)=3

Verified step by step guidance

Identify the given logarithmic equation: \( \log_4(3x+2) = 3 \).

Rewrite the logarithmic equation in its exponential form: \( 3x + 2 = 4^3 \).

Calculate the value of \( 4^3 \) to simplify the equation: \( 3x + 2 = 64 \).

Isolate \( x \) by subtracting 2 from both sides: \( 3x = 62 \).

Solve for \( x \) by dividing both sides by 3: \( x = \frac{62}{3} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined as the power to which a base must be raised to obtain a given number. For example, in the equation log_b(a) = c, b^c = a. Understanding how to manipulate and solve logarithmic equations is crucial for finding the value of x in the given problem.

Domain of Logarithmic Expressions

The domain of a logarithmic expression is the set of all possible input values (x) for which the logarithm is defined. Specifically, the argument of the logarithm must be positive. In the equation log4(3x+2)=3, it is essential to ensure that 3x + 2 > 0 to find valid solutions, as any value outside this domain must be rejected.

Exact vs. Approximate Solutions

In solving logarithmic equations, an exact solution is typically expressed in terms of logarithms or algebraic expressions, while an approximate solution is a numerical value obtained through calculation. For instance, after solving log4(3x+2)=3 for x, one may need to use a calculator to find a decimal approximation, ensuring it is rounded to the specified number of decimal places.