Logarithms Explained Rules & Properties, Condense, Expand, Graphing & Solving Equations Introduction

The video starts by explaining how to evaluate simple logarithms by asking 'base to what power equals the number inside the log'. Examples include log base 2 of 4 (answer: 2) and log base 3 of 9 (answer: 2). It also covers common logarithm (base 10) and the special cases of log of 1 (always 0) and logarithms of negative numbers or zero (do not exist). Fractional and negative answers are explained for cases involving fractions or when the base is larger than the number inside the log.

The change of base formula, log base a of b = log b / log a, is introduced. This formula allows you to convert logarithms to any desired base, including the natural logarithm (ln), which has base 'e'.

The video details key logarithm properties: product rule (log a + log b = log(a*b)), quotient rule (log a - log b = log(a/b)), and power rule (log a^n = n log a). These rules are then applied to condense multiple logarithm expressions into a single one and to expand a single logarithm into multiple terms, including handling fractional exponents and negative signs.

Techniques for simplifying expressions involving natural logarithms (ln) and the base 'e' are demonstrated, such as Ln 1 = 0, Ln e = 1, and e^(Ln x) = x. Similar simplifications are shown for other bases, where base^(log base x) equals x.

The fundamental conversion between logarithmic and exponential forms is explained: log base b of y = x is equivalent to b^x = y. This concept is crucial for solving logarithmic equations.

Several examples of solving logarithmic equations are provided. This includes converting to exponential form, condensing multiple logs, factoring quadratic equations that arise, and identifying extraneous solutions (where the argument of a logarithm would be zero or negative). Equations involving natural logarithms are also solved using the property of base 'e'.

The video demonstrates how to solve exponential equations by converting bases to a common base and equating exponents. For equations where a common base is not easily found, using logarithms (either common log or natural log) on both sides is presented as a method to isolate the variable, often requiring a calculator for the final numerical answer.

The process for determining the domain of logarithmic functions is explained. Since the argument of a logarithm must be greater than zero, an inequality is set up and solved. For quadratic arguments, a sign chart is used to identify the valid intervals for the domain.

Steps to find the inverse function of a given logarithmic function are outlined: replace f(x) with y, switch x and y, and then solve for y. This demonstrates the inverse relationship between logarithmic and exponential functions.

Detailed instructions are given for graphing exponential functions. This involves selecting key x-values by setting the exponent to 0 and 1, calculating corresponding y-values, and identifying the horizontal asymptote. Different transformations based on positive/negative signs in front of x and the base are illustrated, affecting the quadrant towards which the function extends.

The process for graphing logarithmic functions is explained. This involves setting the argument of the log to 0 (for the vertical asymptote), 1, and the base of the logarithm to find key points. The domain and range for logarithmic functions are discussed. The video concludes by demonstrating how to graph the inverse of a logarithmic function, which is an exponential function, highlighting their symmetry across the line y=x.