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Chapter 1: Problem 7

Explain the meaning of \(\log _{b} x\).

Short Answer

Expert verified
In summary, the expression \(\log_bx\) represents the exponent to which the base b must be raised to obtain the number x. It is an inverse operation of exponentiation and can be understood by converting it into an exponential equation: if \(y = \log_bx\), then \(b^y = x\).

Step by step solution

01

Understand Logarithms

Logarithms are the inverse operation of exponentiation. That is, a logarithm gives the answer to the question, "To what exponent must the base be raised in order to get the desired number?" The logarithm of a number x, with respect to a certain base b, is denoted as \(\log_bx\).
02

Interpret \(\log_bx\)

To interpret \(\log_bx\), we can say that it represents the exponent to which the base b must be raised to obtain the number x. In other words, if \(y = \log_bx\), then \(b^y = x\).
03

Convert Logarithmic Equation to Exponential Equation

To better understand and explain the meaning of \(\log_bx\), let's convert it into an exponential equation. Let y = \(\log_bx\). From the definition of logarithms, this means: \(b^y = x\) Now, we can see that \(\log_bx = y\) means that \(b^y = x\), which clearly demonstrates the relationship between the base, exponent, and the original number x in logarithmic form.

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