Skip to main content
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 35
Chapter 5, Problem 35

Solve each equation. log9 x = 5/2

Verified step by step guidance
1
Recognize that the equation is given in logarithmic form: \(\log_{9} x = \frac{5}{2}\). This means the logarithm base is 9, and the logarithm of \(x\) equals \(\frac{5}{2}\).
Recall the definition of logarithm: \(\log_{a} b = c\) means \(a^{c} = b\). Apply this to rewrite the equation as an exponential equation: \(9^{\frac{5}{2}} = x\).
Express the base 9 as a power of a smaller base if possible. Since \(9 = 3^{2}\), rewrite \(9^{\frac{5}{2}}\) as \((3^{2})^{\frac{5}{2}}\).
Use the power of a power property: \((a^{m})^{n} = a^{m \times n}\). So, \((3^{2})^{\frac{5}{2}} = 3^{2 \times \frac{5}{2}}\).
Simplify the exponent multiplication: \(2 \times \frac{5}{2} = 5\). Therefore, \(x = 3^{5}\). This expresses the solution for \(x\) in exponential form.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It answers the question: to what power must the base be raised to produce a given number? For example, log₉(x) = y means 9 raised to y equals x.
Recommended video:
5:26
Graphs of Logarithmic Functions

Properties of Logarithms

Understanding properties like the definition of logarithms and how to rewrite logarithmic equations in exponential form is essential. For instance, logₐ(b) = c can be rewritten as a^c = b, which helps in solving equations.
Recommended video:
5:36
Change of Base Property

Solving Exponential Equations

Once the logarithmic equation is converted to exponential form, solving involves applying exponent rules and algebraic manipulation to isolate the variable. This step is crucial to find the exact value of x.
Recommended video:
5:47
Solving Exponential Equations Using Logs
Related Practice
Textbook Question

Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Determine whether the wetland is a rich fen, a poor fen, or a bog. 2.49×\(\times\)10-5

Textbook Question

Graph each function. ƒ(x) = 4-x

Textbook Question

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. e2x - 6ex + 8 = 0

Textbook Question

Graph each function. ƒ(x)=(1/6)−xƒ(x)=(1/6)^{-x}

5
views
Textbook Question

Find the [H3O+] for each substance with the given pH. Write answers in scientific notation to the nearest tenth. beer, 4.8

Textbook Question

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. 2e2x + ex = 6