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Chapter 4: Problem 123

Explain why \(\log _{a} x\) is defined only for \(01\).

Short Answer

Expert verified
The logarithm function \(\log _{a} x\) is only defined for 01 because a logarithm is the inverse operation to exponentiation. For a>1 or 0<a<1, the function \(a^x\) covers all positive real numbers, allowing the logarithm to be defined. If 'a' equals 1, the function \(a^x\) provides the same value for all 'x', therefore the logarithm is not defined.

Step by step solution

01

Understanding the definition of a logarithm

By definition, the function \(\log _{a} x\) represents the solution to the equation \(a^y = x\), where 'a' is the base, 'x' is the argument and 'y' is the value of the function. 'a' must be a positive real number, but it can not be equal to 1.
02

Evaluating for 0

For 0<a<1, as 'x' gets larger, \(a^x\) gets smaller. This is because raising a number between 0 and 1 to a larger power makes the number decrease. Therefore, \(\log _{a} x\) for 0<a<1 always exists.
03

Evaluating for a>1

For a>1, as 'x' gets larger, \(a^x\) also gets larger. Therefore, \(\log _{a} x\) for a>1 always exists.
04

Understanding the exception

However, if 'a' equals 1, the function \(a^y = x\) would provide the same value for all 'x'. This means the inverse function, the logarithm, isn't defined. Therefore, 'a' can not be 1.

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Most popular questions from this chapter

Use the regression feature of a graphing utility to find a logarithmic model \(y=a+b \ln x\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(3,14.6),(6,11.0),(9,9.0),(12,7.6),(15,6.5)$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x^{2} e^{2 x}+2 x e^{2 x}=0$$

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\ln (x+2)-3^{x-2}+10=5$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1-\ln x}{x^{2}}=0$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln 2 x=1.5$$
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