Logarithmic Functions

1. Consider the exponetial equation. Find (a) the exact solution and (b) an approximation to the solution rounded six decimal places.   solution

(a) $4^{2x-1}=5$   (b) $2e^{5x}=7$   (c) $5(1+10^{8x})=11$   (d) $2(6+8^{x+1})=100$

2. Consider the Logarithmic equation. Find (a) the exact solution and (b) an approximation to the solution rounded six decimal places.   solution

(a) $\ln(3+x)=1$   (b) $\log(3x+3)=1$   (c) $3-\log(5-x)=2$

3. Solve the following Logarithmic equations.  solution

(a) \(\log_2 (3x-1) = 3 \)   (b) \(\log_5(3x-2)=2 \)     (c) \( \log (x^2-7x+11) = 0 \)    (d) \( \displaystyle{\log_3\left( \frac{x+1}{x-2}\right) = 1}\)

4. Solve the following Exponential equations.   solution

(a) \(\displaystyle{ 8^{x + 3} = 16^{x - 1} }\)    (b) \(\displaystyle{ \frac{8}{27} = \left(\frac{3}{2}\right)^x }\)  (c) \(\displaystyle{ (\sqrt 2)^{2 - 3x} = \frac{1}{8} }\)   (d) \(\displaystyle{ 3^{x^2 - 5x - 12} = 9 }\)   (e) \(\displaystyle{ 5^{x^2 - 3x - 35} = \left( \frac{1}{5} \right)^x }\)