Logarithmic and Exponential Forms: Conversions, Evaluations, and Properties

Exponential to Logarithmic Form
- The general form is $b^y = x$ , where:
 - $b$ is the base.
 - $y$ is the exponent.
 - $x$ is the result.
- To convert, it is essential to correctly identify $b$ , $y$ , and $x$ in the exponential equation.
- The logarithmic form is written as $\log_b(x) = y$ .
- Example:
 - Given $b^3 = 27$ , then $\logb(27) = 3$ or $\logb27 = 3$ .
Logarithmic to Exponential Form
- The logarithmic form is generally expressed as $\log_b(x) = y$ .
  - Here, $b$ is the base, $x$ is the argument of the logarithm, and $y$ is the result.
- To convert to exponential form, rewrite as $b^y = x$ .
- Example:
  - Given $\log_2(25) = y$ , convert to $2^y = 25$ .

Using Exponential Form
- If a logarithmic expression is given, set it equal to $y$ . For example, $\log_{10}(100) = y$ .
- Convert the logarithmic form to exponential form to solve for $y$ .
Example: Evaluating $\log_{10}(100)$
- Set $\log_{10}(100) = y$ .
- Convert to exponential form: $10^y = 100$ .
- Express both sides with the same base: $10^y = 10^2$ .
- According to the one-to-one property, if the bases are equal, the exponents must be equal: $y = 2$ .
- Therefore, $\log_{10}(100) = 2$ .
Example: Evaluating $\log_5(\frac{1}{125})$
- Set $\log_5(\frac{1}{125}) = y$ .
- Convert to exponential form: $5^y = \frac{1}{125}$ .
- Rewrite the fraction using a negative exponent: $5^y = 125^{-1}$ .
- Express 125 as a power of 5: $5^y = (5^3)^{-1}$ .
- Simplify the exponent: $5^y = 5^{-3}$ .
- Equate the exponents: $y = -3$ .

These include properties that can simplify logarithmic expressions.
Property 1: $\log_b(b) = 1$
- If the base and the argument are the same, the logarithm equals 1.
- Example: $\log_9(9) = y$ implies $9^y = 9$ , so $y = 1$ .
Property 2: $\log_b(1) = 0$
- The logarithm of 1 to any base is 0.
- Example: $\log_e(1) = y$ implies $e^y = 1$ , so $y = 0$ .
Property 3: $b^{\log_b(x)} = x$
- If you raise a base to a logarithm with the same base, it equals the argument.
- Example: Evaluate $7^{\log_7(8)}$ .
  - Since the bases are the same, the expression equals 8.
Property 4: If $\logb(A) = \logb(B)$ , then $A = B$
- If two logarithms with the same base are equal, their arguments are equal.
- Example: Given $\log3(x) = \log3(17)$ , then $x = 17$ .

Natural Logarithm
- The natural logarithm is log base $e$ , where $e$ is Euler's number (approximately 2.71828).
- It is written as $\ln(x)$ , which is equivalent to $\log_e(x)$ .

General Forms
- Exponential function: $f(x) = b^x$ .
- Logarithmic function: $g(x) = \log_b(x)$ .
Inverse Relationship
- Logarithmic and exponential functions are inverses of each other.
- To graph, one can find points on the exponential function and then switch the $x$ and $y$ values to graph the logarithmic function.
Asymptotes
- Exponential functions have a horizontal asymptote.
- Logarithmic functions have a vertical asymptote.
- For $f(x) = 3^x$ , the horizontal asymptote is at $y = 0$ .
- For $g(x) = \log_3(x)$ , the vertical asymptote is at $x = 0$ .
Graphing Process
- Create a table of values for the exponential function $f(x) = 3^x$ using easy-to-calculate values for $x$ , such as -2, -1, 0, 1, and 2.
- Calculate corresponding $y$ values: $3^{-2} = \frac{1}{9}$ , $3^{-1} = \frac{1}{3}$ , $3^0 = 1$ , $3^1 = 3$ , $3^2 = 9$ .
- To graph the logarithmic function $g(x) = \log_3(x)$ , switch the $x$ and $y$ values from the exponential function's table.
- The points for $g(x)$ are ( $\frac{1}{9}$ , -2), ( $\frac{1}{3}$ , -1), (1, 0), (3, 1), and (9, 2).

When you are stuck on solving a logarithmic form, convert to exponential form.
Memorizing logarithmic properties is helpful, but understanding how to derive them using exponential forms is more valuable.
When graphing logarithmic and exponential functions, recognize their inverse relationship to simplify the process.