/*! 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 42 Evaluate each expression without... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 42

Evaluate each expression without using a calculator. $$7^{\log _{7} 23}$$

Short Answer

Expert verified
The expression \(7^{\log _{7} 23}\) evaluates to 23.

Step by step solution

01

Understand the expression

In the expression \(7^{\log _{7} 23}\), the base of the exponent and the base of the logarithm are the same. The logarithm \(\log _{7} 23\) is simply asking 'to what power must we raise 7 to obtain 23?' And that is the power to which 7 is being raised when we evaluate \(7^{\log _{7} 23}\).
02

Apply the Exponent and Logarithm Identity

The identity \(a^{\log _{a} x} = x\) tells us that any base 'a' raised to the power of its logarithm in the same base of what we denote as 'x', will always equals to 'x'. This is true in our case since the base of the exponent and the base of the logarithm are the same, which is 7; so \(7^{\log _{7} 23} = 23\).

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Most popular questions from this chapter

The bar graph indicates that the percentage of fi rst-year college students expressing antifeminist views declined after 1970. CAN'T COPY THE GRAPH The function $$f(x)=-4.82 \ln x+32.5$$ models the percentage of first-year college women, \(f(x)\) expressing antifeminist views (by agreeing with the statement) \(x\) years after 1969 a. Use the function to find the percentage of first-year college women expressing antifeminist views in 2008 . Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph? By how much? b. Use the function to project the percentage of first-year college women who will express antifeminist views in \(2015 .\) Round to one decimal place.

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Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$\log _{3}(4 x-7)=2$$

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 . \quad\) Use \(\quad\) a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.
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