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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 39
Chapter 5, Problem 39

Evaluate each expression without using a calculator. log5 57

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1
Recognize that the expression is \( \log_5 5^7 \), which is a logarithm with base 5 of \( 5^7 \).
Recall the logarithmic identity: \( \log_b b^x = x \), which means the logarithm of a base raised to a power is just the exponent.
Apply this identity directly to the expression: \( \log_5 5^7 = 7 \).
Therefore, the value of \( \log_5 5^7 \) simplifies to the exponent 7 without further calculation.
This shows how logarithms and exponents are inverse operations, making such expressions straightforward to evaluate.

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Key Concepts

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

Logarithm Definition

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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Logarithm of a Power

The logarithm of a number raised to an exponent can be simplified using the rule log_b(a^c) = c * log_b(a). This property allows you to bring the exponent in front as a multiplier, simplifying the evaluation process.
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Logarithm of the Base

When the argument of a logarithm is the same as its base, log_b(b) equals 1 because b^1 = b. This fact helps simplify expressions like log5(5^7) by reducing the inner logarithm to a simple exponent.
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Related Practice
Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log 5 + log 2

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Textbook Question

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = ex-1+2

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Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(10x21−x37(x+1)2)\(\log\) \(\left\)( \(\frac{10x^2 \sqrt[3]{1 - x}\)}{7(x + 1)^2} \(\right\))

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Textbook Question

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = ln x and g(x) = - ln (2x)

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Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 70.3x=813

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Textbook Question

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e-x

1039
views