9.3 Homework Problems: Functions of Logarithms and Solving Logarithmic Equations

1. Write the logarithmic expression in a different form (if not possible, write N/A)

a. log [latex]\left(\frac{x}{y^3}\right)[/latex]

b. log (a) + log (b)

c. – ln ([latex]\frac{1}{x}[/latex])

d. ln (a + b)

2. Find the value of each expression

a. log₂ (32)

b. log₅[latex]\left(\frac{1}{125}\right)[/latex]

c. [latex]\log_2\sqrt8[/latex]

d. [latex]3^{\log_5(5)}[/latex]

3. Use the properties of logs to fully expand each expression

a. log_b (5x⁴)

b. log_b (x²y)⁴

c. log₅ (xy)^-2

d. log₅(21x²y^2/3)

e. [latex]log\ \frac{\sqrt{x^5}}{(x+4)^3}[/latex]

4. Use the properties of logs to write each expression as a single log

a. 2 log_b (3) + log_b (x) – 2 log_b (5)

b. log_b (x) – 2 log_b (y) – 2 log_b (z)

c. 3 log₅ (y) – [latex]\frac{1}{2}[/latex]log₅ (x)

d. –[latex]\frac{2}{3}[/latex]log₂ (x) – [latex]\frac{1}{3}[/latex]log₂ (y) + [latex]\frac{2}{3}[/latex]log₂ (z)

5. Solve for x (round to the nearest hundredth)

a. log (225) ÷ log (15) = log (x)

b. log (x) + log (x-1) = log (4x)

c. log₂ (x-2) + log₂ (x+1) = 2

d. e ^2x – 2e ^2x = -15

e. ln (x) = ln (5) – ln (2)

f. 3^{2x + 1} = 2^{x – 2}

g. ln ((3)(x+1)) = 4

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