/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Evaluate the indicated expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 22

Evaluate the indicated expression. Do not use a calculator for these exercises. $$ \log _{8} 2^{6.3} $$

Short Answer

Expert verified
The short version of the answer is: \(\log_{8}{2^{6.3}} = 2.1\).

Step by step solution

01

Identify the base and exponent

The given logarithmic expression can be written as \(\log_{8}{a}\), where \(a = 2^{6.3}\). The base of the logarithm is 8 and the expression inside the logarithm is \(2^{6.3}\).
02

Express the base as a power of 2

Since 8 can be expressed as a power of 2, we rewrite the base as \(8 = 2^3\).
03

Use the change of base formula

The change of base formula states that \(\log_{b}{x} = \frac{\log_{c}{x}}{\log_{c}{b}}\), where c can be any positive number different than 1. In our case, let's use base 2: \[\log_{2^3}{2^{6.3}} = \frac{\log_{2}{2^{6.3}}}{\log_{2}{2^3}}\]
04

Simplify the expression using logarithm properties

Using logarithm properties: 1. \(\log_{a}{a^x} = x\) 2. \(\frac{\log_{a}{x}}{\log_{a}{y}} = \log_{y}{x}\) Applying property 1 to both numerator and denominator, we get: \[\frac{6.3}{3}\]
05

Divide and find the final result

Now, just divide the numbers: \[\frac{6.3}{3} = 2.1\] So, the value of the given expression is: \[\log_{8}{2^{6.3}} = 2.1\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find all numbers \(x\) such that the indicated equation holds. \(10^{2 x}-3 \cdot 10^{x}=18\)

For each of the functions \(f\); (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part ( \(c\) ) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I .\) (Recall that \(I\) is the function defined by \(I(x)=x .)\) \(f(x)=4-2 e^{8 x}\)

For each of the functions \(f\); (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part ( \(c\) ) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I .\) (Recall that \(I\) is the function defined by \(I(x)=x .)\) \(f(x)=5 e^{9 x}\)

Suppose \(f\) is a function with exponential growth. Show that there is a number \(b>1\) such that $$ f(x+1)=b f(x) $$ for every \(x\).

Find all numbers \(x\) that satisfy the given equation. .\(e^{2 x}-4 e^{x}=12\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.