Problem 105
Question
Without using a calculator, find the exact value of $$\frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001}$$
Step-by-Step Solution
Verified Answer
The exact value of the expression is \( \frac{2}{3} \).
1Step 1: Simplify the Numerator
Use the properties of logarithms to simplify the expressions in the numerator. The first term, \( \log_3(81) \), simplifies to 4 since \( 3^4 = 81 \). In the second term, \( \log_{\pi}(1) \), this equals 0 because the log of 1 to any base equals zero. The difference between these two yields \( 4 - 0 = 4 \). So, the numerator equals 4.
2Step 2: Simplify the Denominator
Use the properties of logarithms to simplify the expressions in the denominator. The first term, \( \log_{2 \sqrt{2}}(8) \), simplifies to 3 since \( (2 \sqrt{2})^3 = 8 \). The second term, \( \log(0.001) \), equals -3 as the log of \( 0.001 \) to base 10 equals to -3. The difference between these two yields \( 3 - (-3) = 6 \). So, the denominator is 6.
3Step 3: Final Division
Perform the final division: \( \frac{4}{6} \) simplifies to \( \frac{2}{3} \).
Other exercises in this chapter
Problem 104
Which one of the following is true? a. \(\frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4}\) b. \(\log (-100)=-2\) c. The domain of \(f(x)=\log _{2} x\) is \((-\infty
View solution Problem 105
If \(f(x)=\log _{\phi} x,\) show that \(\frac{f(x+h)-f(x)}{h}=\log _{b}\left(1+\frac{h}{x}\right)^{1 / h}, h \neq 0\)
View solution Problem 106
Solve for \(x: \log _{4}\left[\log _{3}\left(\log _{2} x\right)\right]=0.\)
View solution Problem 107
Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40.\)
View solution