Logarithms Review

Evaluating Logarithms

• To evaluate logarithms, set the expression equal to $x$ and rewrite in exponential form.
• Solve for $x$ by recognizing the relationship between the base and the result.

Examples of Evaluating Logarithms

• $\log_{6} 36 = x \Rightarrow 6^x = 36 \Rightarrow x = 2$ because $6^2 = 36$.
• $\log_{4} \frac{1}{64} = x \Rightarrow 4^x = \frac{1}{64} \Rightarrow x = -3$ because $4^{-3} = \frac{1}{64}$.
• $\log_{5} 5 = x \Rightarrow 5^x = 5 \Rightarrow x = 1$.
• $\log_{2} 32 = x \Rightarrow 2^x = 32 \Rightarrow x = 5$ because $2^5 = 32$.
• $\log_{9} 1 = x \Rightarrow 9^x = 1 \Rightarrow x = 0$ because any number raised to the power of 0 equals 1.
• $\log_{8} \frac{1}{64} = x \Rightarrow 8^x = \frac{1}{64} \Rightarrow x = -2$ because $8^{-2} = \frac{1}{64}$.
• $\log_{6} \frac{1}{6} = y \Rightarrow 6^y = \frac{1}{6} \Rightarrow y = -1$ because $6^{-1} = \frac{1}{6}$.
• If there is no base number, it's assumed to be base 10.
• $\log 100 = x \Rightarrow 10^x = 100 \Rightarrow x = 2$.

Condensing Logarithmic Expressions

• Condensing logarithmic expressions involves using the properties of logarithms to combine multiple logarithms into a single logarithm.
• Subtraction condenses to division (quotient rule).
• Addition condenses to multiplication (product rule).
• A number in front of a logarithm becomes an exponent (power rule).

Subtraction to Division

• When logarithms are subtracted, condense them by dividing their arguments.
• $\log 2 - \log 5 = \log \frac{2}{5}$

Addition to Multiplication

• When logarithms are added, condense them by multiplying their arguments.
• $\log 7 + \log x = \log 77 \Rightarrow 7 * x = 77 \Rightarrow x = 11$

Number in Front to Exponent

• A number multiplied by a logarithm becomes the exponent of the logarithm's argument.
• $-5 \log<em>{9} 2 = \log</em>{9} 2^{-5} = \log_{9} \frac{1}{32}$

More Examples of Condensing

• $\log 11 - \log 8 = \log \frac{11}{8}$
• $\log 5 + \log x = \log 45 \Rightarrow 5 * x = 45 \Rightarrow x = 9$
• $-2 \log<em>{5} x \Rightarrow \log</em>5 x^{-2}$
• $\frac{1}{4} = x^{-2} \Rightarrow x = 2$
• $\log 2 + \log 5 = \log 10$
• $\log 3 - \log x = \log 3 \Rightarrow x = x$
• $25 = 5^x \Rightarrow x = 2$
• $\log 9 + \log 5 = \log 45$
• $\log 10 - \log 2 = \log 5$
• $\log_2 5^{-2} = x \Rightarrow x = \frac{1}{25}$

Expanding Logarithmic Expressions

• Expanding logarithms involves using the properties of logarithms to break down a single logarithm into multiple logarithms.
• Multiplication expands to addition.
• Division expands to subtraction.
• Exponents become coefficients.

Multiplication to Addition

• $\log (z^3 * x) = 3 \log z + \log x$
• $\log (y * z^2) = \log y + 2 \log z$

Division to Subtraction

• $\log(\frac{x}{y^7}) = \log x - 7 \log y$

Summary

• Evaluating expressions will not be part of the video assessment but is on topics in the assignment.
• Expanding, where they are expanded and you're condensing them, will be a question on your video assessment.