Problem 123
Question
Explain why \(\log _{a} x\) is defined only for \(0
Step-by-Step Solution
Verified Answer
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
1Step 1: 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.
2Step 2: Evaluating for 0<a<1
For 0
3Step 3: 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.
4Step 4: 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.
Other exercises in this chapter
Problem 122
Prove that \(\frac{\log _{a} x}{\log _{a / b} x}=1+\log _{a} \frac{1}{b}\).
View solution Problem 123
Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\ln (x+2)-3^{x-2}+10=5$$
View solution Problem 123
Simplify the expression.$$\left(64 x^{3} y^{4}\right)^{-3}\left(8 x^{3} y^{2}\right)^{4}$$
View solution Problem 124
Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\ln x^{2}-e^{x}=-3-\ln x^{2}$$
View solution