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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 83
Chapter 5, Problem 83

Evaluate or simplify each expression without using a calculator. log 107

Verified step by step guidance
1
Recall the definition of logarithms: \(\log_b(a)\) asks the question, "To what power must the base \(b\) be raised to get \(a\)?"
In this problem, the base of the logarithm is 10, and the argument is \$10^7$, so we are looking for the exponent $x$ such that \(10^x = 10^7\).
Since the bases on both sides of the equation are the same (base 10), the exponents must be equal. Therefore, \(x = 7\).
Thus, \(\log 10^{7} = 7\) because the logarithm of a power of 10 with base 10 is simply the exponent.
This simplification uses the logarithmic identity: \(\log_b(b^k) = k\) for any positive base \(b \neq 1\) and any real number \(k\).

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Key Concepts

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

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential for evaluating logarithmic expressions.
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Logarithms Introduction

Logarithm of a Power

The logarithm of a number raised to an exponent can be simplified using the rule log_b(a^n) = n * log_b(a). This property allows you to bring the exponent in front as a multiplier, simplifying calculations without a calculator.
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Common Logarithms (Base 10)

Common logarithms have base 10, written as log or log_10. Since log_10(10) = 1, powers of 10 simplify easily: log_10(10^n) = n. Recognizing this helps quickly evaluate expressions like log 10^7.
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Related Practice
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. log(x+4)−log 2=log(5x+1)

Textbook Question

In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C.

logb (3/2)

Textbook Question

Use the formula for continuous compounding to solve Exercises 84–85. How long, to the nearest tenth of a year, will it take \$50,000 to triple in value at an annual rate of 7.5% compounded continuously?

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Textbook Question

Expand: log8(4√x64y3)log_8\(\left\)(\(\frac{4\surd x}{64y3}\)\(\right\))

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Textbook Question

In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C.

logb 8

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Textbook Question

Use the formula for continuous compounding to solve Exercises 84–85. What annual rate, to the nearest percent, is required for an investment subject to continuous compounding to triple in 5 years?

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