Logarithmic Form
The logarithmic form of an equation is written as:
$$\log_b a = x$$
where:
- \$b\$ is the base of the logarithm.
- \$a\$ is the argument of the logarithm.
- \$x\$ is the exponent to which the base must be raised to obtain \$a\$.
Exponential Form
The exponential form of an equation is written as:
$$b^x = a$$
where:
- \$b\$ is the base.
- \$x\$ is the exponent.
- \$a\$ is the result of raising the base to the power of the exponent.
Relationship Between Logarithmic and Exponential Forms
The logarithmic and exponential forms are equivalent and can be converted from one to the other:
- If \$\log_b a = x\$, then \$b^x = a\$.
- If \$b^x = a\$, then \$\log_b a = x\$.
Important Properties of Logarithms
- Logarithm of 1: \$\log_b 1 = 0\$ for any base \$b > 0\$ and \$b \neq 1\$.
- Logarithm of the Base: \$\log_b b = 1\$ for any base \$b > 0\$ and \$b \neq 1\$.
- Logarithm of a Power: \$\log_b (b^x) = x\$ for any base \$b > 0\$ and \$b \neq 1\$.
Solving Exponential Equations Using Logarithms
Logarithms can be used to solve exponential equations by converting them into logarithmic form.
Logarithms with Different Bases
Most calculators have buttons for \$\log_{10}\$ (common logarithm) and \$\log_e\$ (natural logarithm, denoted as \$\ln\$). To calculate logarithms with other bases, use the change of base formula:
$$\log_b a = \frac{\log_c a}{\log_c b}$$
where \$c\$ is any positive number (usually 10 or \$e\$).
Graphs of Logarithmic Functions
The graph of a logarithmic function \$y = \log_b x\$ has the following characteristics:
- Domain: \$x > 0\$
- Range: All real numbers
- Vertical Asymptote: \$x = 0\$
- Intercept: \$(1, 0)\$
- Increasing: If \$b > 1\$
- Decreasing: If \$0 < b < 1\$
Applications of Logarithms
Logarithms have many applications in real-world situations, including:
- Exponential Growth and Decay: Logarithms are used to model and solve problems involving exponential growth and decay, such as population growth, radioactive decay, and compound interest.
- Sound Intensity: The decibel scale, which measures sound intensity, is a logarithmic scale.
- Earthquake Magnitude: The Richter scale, which measures earthquake magnitude, is a logarithmic scale.
Problem: A population of bacteria doubles every 3 hours. If the initial population is 100, how long will it take for the population to reach 800?
- The population can be modeled by the equation \$P(t) = 100 \cdot 2^{t/3}\$, where \$t\$ is the time in hours.
- We want to find \$t\$ such that \$P(t) = 800\$.
- Set up the equation: \$100 \cdot 2^{t/3} = 800\$.
- Divide both sides by 100: \$2^{t/3} = 8\$.
- Convert to logarithmic form: \$\log_2 8 = \frac{t}{3}\$.
- Since \$2^3 = 8\$, we have \$\log_2 8 = 3\$.
- Solve for \$t\$: \$t = 3 \cdot 3 = 9\$.
- Therefore, it will take 9 hours for the population to reach 800.
