Logarithmic Functions

1. 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

2.  Write the following in expanded form. Where possible, evaluate the log expressions.  solution

(a) \(\displaystyle{\ln [x(x+5)]} \)       (b) \(\displaystyle{ \log_2 \frac{x^4}{8} }\)        (c) \(\displaystyle{ \log_7 \frac{\sqrt 7}{x^3} }\)       (d) \(\displaystyle{ \ln \frac{2e^2}{3}}\)

3. Write the following in condensed form.  solution

(a) \(\displaystyle{ \log x + 2 \log y}\)

(b) \(\displaystyle{ 2\ln x - 5\ln(x^2 + 1) }\)

(c) \(\displaystyle{ 2\ln x - \frac{1}{2} \ln(x^2 + 1) }\)

(d) \(\displaystyle{ 3\log_5(3x + 10) - 2\log_5(x^2 - 4x) + 4\log_5(z)}\)

4. 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