### Exponentials & Logarithms

Package of problems

## Some algebraic problems

1. Evaluate the following. Can you do this without calculating each factor separately?

1. $(\log_3 9)(\log_9 81)$

2. $(\log_3 81)(\log_{81} 9)$

2. Write the following as a single logarithm. What conditions must $x$ and $y$ satisfy?

1. $(\log_{10} 5)(\log_{5} x)$

2. $(\log_{10} x)(\log_{x} y)(\log_{y} 3)$

3. Simplify the following. What conditions must $x$ and $y$ satisfy?

1. $(\log_{x} y)(\log_{y} 10)(\log_{8} x)$

2. $(\log_{x} y)(\log_{8} x)(\log_{y} 8)$

4. The value of $\log_2 3$ is approximately $1.6$. Can you use this to approximate $\log_4 27$? What about $\log_8 27?$ Can you generalise this result?

5. Evaluate

1. $\dfrac{\log_{2} 5}{\log_{2} 25}$

2. $\dfrac{\log_{10} 5}{\log_{10} 3}$

6. Given that $\log_x y=4$ and $\log_y 2 = 9$, evaluate

1. $(\log_x 2)(\log_y x)$

2. $\dfrac{\log_x 2}{\log_y x}$

7. What is $(\log_a b^3)(\log_b a^3)$?

8. What is the relationship between $\log_a b$ and $\log_b a$?

9. Solve $\log_{x} 7 = \log_{7} x$.