key term - Properties of Logarithms

Definition

The properties of logarithms are the fundamental rules that govern the behavior of logarithmic functions. These properties describe how logarithms can be manipulated and used to solve exponential and logarithmic equations, which are the focus of the 6.6 Exponential and Logarithmic Equations topic.

5 Must Know Facts For Your Next Test

1. The product rule for logarithms states that $\log_a (xy) = \log_a x + \log_a y$.
2. The power rule for logarithms states that $\log_a (x^n) = n\log_a x$.
3. The quotient rule for logarithms states that $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$.
4. The logarithm of 1 is always 0, regardless of the base: $\log_a 1 = 0$.
5. The logarithm of the base itself is always 1: $\log_a a = 1$.

Review Questions

• Explain how the properties of logarithms can be used to solve exponential equations.
  • The properties of logarithms, such as the product rule, power rule, and quotient rule, allow us to manipulate exponential equations and rewrite them in logarithmic form. This makes it possible to solve for the unknown variable by applying the inverse relationship between exponential and logarithmic functions. For example, to solve an equation like $2^x = 32$, we can take the logarithm of both sides using the power rule: $\log_2 (2^x) = \log_2 32$, which simplifies to $x = \log_2 32$.
• Describe how the change of base formula for logarithms can be used to convert between different logarithmic bases.
  • The change of base formula, $\log_a x = \frac{\log_b x}{\log_b a}$, allows you to convert a logarithm with one base (base $b$) to a logarithm with a different base (base $a$). This is useful when working with logarithmic expressions that have different bases, as it enables you to standardize the base and perform further manipulations. For instance, if you need to evaluate $\log_3 27$ but only know how to work with common logarithms (base 10), you can use the change of base formula to convert: $\log_3 27 = \frac{\log_{10} 27}{\log_{10} 3}$.
• Analyze how the properties of logarithms can be used to simplify complex logarithmic expressions.
  • The properties of logarithms, such as the product rule, power rule, and quotient rule, allow you to break down and simplify complex logarithmic expressions. By applying these rules strategically, you can combine like terms, eliminate exponents, and rewrite the expression in a more manageable form. For example, to simplify $\log_2 (x^3 \cdot y^2) - \log_2 (x^2)$, you would use the product rule to combine the logs of the numerator, the power rule to simplify the exponents, and the quotient rule to subtract the logs, resulting in $3\log_2 x + 2\log_2 y - 2\log_2 x$.

Related terms

Logarithm:

The logarithm of a number is the exponent to which a fixed base must be raised to get that number.

Exponential Function:

An exponential function is a function in which the variable appears as an exponent, often written in the form $y = a^x$, where $a$ is the base.

Change of Base Formula:

The change of base formula allows you to convert a logarithm with one base to a logarithm with a different base, using the formula $\log_a x = \frac{\log_b x}{\log_b a}$.