Logarithmic Functions

1. Write in logarithmic form.  (a)  \( 2^5=32 \)   (b)  \( \left( \frac{1}{3}\right)^{-2}=9\)   solution

2. Convert to an exponential form. (a) \( \log_3 81 = 4 \)    (b) \(\log_a b = c \)   solution

3. Find the logarithm.  (a) \(\log_5 \frac{1}{125} \)  (b) \(\log_8 2  \)   solution

4. Find the exact value of the logarithm. (a) \(\log_5 \sqrt[3] 5 \)    (b) \(\log_2(\log_3 81) \)    (c) \(\log_5(\log_7 7) \)   solution

5. Solve the following equations. 

(a)  \(\log_2 (3x-1) = 3 \)   solution

(b)  \(\log_5(3x-2)=2 \)  solution

(c)  \( \log (x^2-7x+11) = 0 \)   solution

(d)  \( \displaystyle{\log_3\left( \frac{x+1}{x-2}\right) = 1}\)   solution

6. Describe briefly how the graph of \(f(x)=\log_3(x-2) \) can be obtained from the graph of \( y=\log_3 x\). Then graph the function and find the domain, the vertical asymptote, x-intercept, and y-intercept.   solution

7. Find the domain of \( f(x)= \log \left( \frac{x-5}{x+2} \right) \).   solution