Expand and Condense Logarithmic Functions

1. Given that \(\displaystyle{\log x = 4, \log y = 2},  \log 2 \approx 0.3 ,\) and \( \log 3 \approx 0.48 \), evaluate (without using a calcluator)  solution 

   (a) \(\log 6 \)    (b) \(\log (3xy^2) \)    (c) \(\log (12 xy^2) \)    (d) \(\log \left(\frac{5x}{2 \sqrt[3]y} \right)\)

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. Use a calculator and the base-change formula to find each logarithm to four decimal places.   solution

(a) \(\displaystyle{ \log_8 27 }\)

(b) \(\displaystyle{ \log_2 7 + \log_3 22 }\)

(c) \(\displaystyle{ \log_5 17 - \log_3 5 }\)

5. Evaluate the following (without using a calculator).  solution

(a)  \(\displaystyle{ \log_5 \sqrt 5 }\)      (b) \(\displaystyle{ {\log_{12} 1} }\)

(c) \(\displaystyle{ 5^{\log_5 20 - \log_5 2} }\)   (d) \(\displaystyle{ 7^{2 \log_7 5 + \log_7 2} }\)

(e) \(\displaystyle{ 9^{\log_3 \sqrt 2} }\)    (f) \(\displaystyle{ e^{\frac{1}{3} \ln 8} }\)

6. Suppose that $2500 is invested in an account that pays interest compounded continuously. Find the amount of time that it would take for the account to grow to $5000 at 3.5%.    solution

7. The decay rate of a Tritium is 5.36% per year. What is its half-life?   solution