Question

2 use the properties of logarithms to rewrite each logarithmic expression as a single logarithm.
a ( 2 log_{5} 3 - log_{5} y )
b ( 7 ln x + ln 8 - 3 ln y )
c ( 2(log 5 + log m) - log (m^{3}) )
d ( 8 log_{2} x - 3(log_{2} y + 2 log_{2} x) )

Explanation:

Part (a)

Step1: Apply Power Rule

The power rule of logarithms states that \( n\log_b a=\log_b a^n \). So, for \( 2\log_5 3 \), we can rewrite it as \( \log_5 3^2=\log_5 9 \).
Now the expression becomes \( \log_5 9 - \log_5 y \).

Step2: Apply Quotient Rule

The quotient rule of logarithms states that \( \log_b a-\log_b c = \log_b\frac{a}{c} \).
Applying this rule to \( \log_5 9 - \log_5 y \), we get \( \log_5\frac{9}{y} \).

Step1: Apply Power Rule to each term

For \( 7\ln x \), using \( n\ln a=\ln a^n \), we get \( \ln x^7 \).
For \( 3\ln y \), we get \( \ln y^3 \).
The expression now is \( \ln x^7+\ln 8 - \ln y^3 \).

Step2: Apply Product Rule

The product rule of logarithms states that \( \ln a+\ln b=\ln(ab) \).
Applying this to \( \ln x^7+\ln 8 \), we get \( \ln(8x^7) \).
Now the expression is \( \ln(8x^7)-\ln y^3 \).

Step3: Apply Quotient Rule

Using \( \ln a - \ln b=\ln\frac{a}{b} \), we get \( \ln\frac{8x^7}{y^3} \).

Step1: Apply Product Rule inside the parentheses

The product rule \( \log a+\log b = \log(ab) \) to \( \log 5+\log m \), we get \( \log(5m) \).
Now the expression is \( 2\log(5m)-\log(m^3) \).

Step2: Apply Power Rule

Using \( n\log a=\log a^n \) to \( 2\log(5m) \), we get \( \log((5m)^2)=\log(25m^2) \).
The expression becomes \( \log(25m^2)-\log(m^3) \).

Step3: Apply Quotient Rule

Using \( \log a-\log b=\log\frac{a}{b} \), we get \( \log\frac{25m^2}{m^3} \).

Step4: Simplify the fraction

Simplify \( \frac{25m^2}{m^3}=\frac{25}{m} \) (for \( m
eq0 \)). So the expression is \( \log\frac{25}{m} \).

Answer:

\( \log_5\frac{9}{y} \)

Part (b)