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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q. 10.40 (page 1023)

Find the value of $x$ .
$a) \log_{x} (81) = 2 b) \log_{3} (x) = 5 c) \log_{\frac{1}{3}} (\frac{1}{27}) = x$

Short Answer

Expert verified

Part (a) The value of $x$ is $9$ .

Part (b) The value of $x$ is $243$ .

Part (c) The value of $x$ is $3$ .

Step by step solution

01

Step 1. Given information.

The given expressions are,

$a) \log_{x} (81) = 2 b) \log_{3} (x) = 5 c) \log_{\frac{1}{3}} (\frac{1}{27}) = x$

02

Solving part (a).

Here, we will convert the given expression to the exponential form and solve the quadratic equation.

$\log_{x} (81) = 2 x^{2} = 81 x = 9, - 9$

The base of a logarithmic function cannot be negative hence the value of $x$ is $9$ .

03

Step 3. Solving part (b).

Now, we will convert the given expression to exponential form and simplify.

$\log_{3} (x) = 5 x = 3^{5} x = 243$

Therefore the value of $x$ is $243$ .

04

Step 4. Solving part (c).

Now, we will convert the given expression to exponential form and simplify.

$\log_{\frac{1}{3}} (\frac{1}{27}) = x {(\frac{1}{3})}^{x} = \frac{1}{27} {(\frac{1}{3})}^{x} = {(\frac{1}{3})}^{3} x = 3$

Therefore the value of $x$ is $3$ .

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Most popular questions from this chapter

In the following exercises, graph each exponential function.

$g (x) = 7^{x}$

In the following exercises, solve each logarithmic equation.

$\log_{a} 16 = 2$

On the same coordinate system, graph:

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$g (x) = 3^{x} + 2$

In the following exercises, solve each equation.

$\frac{e^{x^{2}}}{e^{3}} = e^{2 x}$

Convert from exponential to logarithmic form:

$4^{- 3} = \frac{1}{64}$ .

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