A detailed look at the essential properties of logarithms used to expand, condense, and evaluate logarithmic expressions. Mastering these rules is crucial for solving logarithmic equations.
The logarithm of a product is the sum of the logarithms of its factors. \(\log_b(MN) = \log_b(M) + \log_b(N)\).
State the Quotient Rule for logarithms.
The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. \(\log_b(M/N) = \log_b(M) - \log_b(N)\).
State the Power Rule for logarithms.
The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. \(\log_b(M^p) = p \cdot \log_b(M)\).
Use the properties of logarithms to expand the expression $$\ln\left(\frac{x^3}{\sqrt{y}}\right)$$
First apply the Quotient Rule, then the Power Rule:$$\ln(x^3) - \ln(y^{1/2}) = 3\ln(x) - \frac{1}{2}\ln(y)$$
Use the properties of logarithms to condense the expression$$2\log(a) + 3\log(b) - \log(c)$$
First apply the Power Rule, then the Product and Quotient Rules:$$\log(a^2) + \log(b^3) - \log(c) = \log\left(\frac{a^2b^3}{c}\right)$$
What is the value of \(\log_b(1)\) for any valid base \(b\)?
\(\log_b(1) = 0\), because \(b^0 = 1\).
What is the value of \(\log_b(b)\) for any valid base \(b\)?
\(\log_b(b) = 1\), because \(b^1 = b\).
State the Change of Base formula for logarithms.
For any positive numbers \(M\), \(b\), and \(c\) (with \(b \neq 1\) and \(c \neq 1\)), \(\log_b(M) = \frac{\log_c(M)}{\log_c(b)}\).
Why is the Change of Base formula useful, especially when using a calculator?
Most calculators only have buttons for the common logarithm (\(\log\), base \(10\)) and the natural logarithm (\(\ln\), base \(e\)). The formula allows you to evaluate logarithms of any base using one of these two.
Write an expression to evaluate \(\log_7(50)\) using a calculator with only a \(\log\) button.
Using the Change of Base formula with base \(10\): \(\log_7(50) = \frac{\log(50)}{\log(7)}\).
Write an expression to evaluate \(\log_7(50)\) using a calculator with only an \(\ln\) button.
Using the Change of Base formula with base \(e\):$$\log_7(50) = \frac{\ln(50)}{\ln(7)}$$
Simplify the expression \(\log_3(3^{x+2})\) using inverse properties.
The expression simplifies to \(x+2\), because \(\log_b(b^y) = y\).
Simplify the expression \(e^{\ln(5x)}\) using inverse properties.
The expression simplifies to \(5x\), because \(b^{\log_b(y)} = y\).
