Exponentials and Logarithms: A Comprehensive Guide

$x = a^n \iff \log_a x = n$

In this notation, '$a$' is known as the base of the logarithm. Essentially, a logarithm asks the question: "To what power must I raise the base $a$ to get the value $x$?".

Examples of conversion:

• $3^2 = 9$ can be written as $\log_3 9 = 2$.
• $10^3 = 1000$ can be written as $\log_{10} 1000 = 3$.

Calculator Tip: Most scientific calculators have a "log" button that defaults to base 10. If you type $\log 20$, the calculator is finding the power to which 10 must be raised to get 20 ($\log_{10} 20 \approx 1.30$).

The Three Laws of Logarithms

To manipulate complex mathematical expressions, we use three fundamental laws. These laws allow us to expand a single logarithm into multiple terms or condense multiple terms into one.

1. The Multiplication Law

The logarithm of a product is the sum of the logarithms of the individual factors: $\log_a xy = \log_a x + \log_a y$ This is derived from the exponent rule $a^b \times a^c = a^{b+c}$. Since multiplying numbers with the same base results in adding their exponents, the logarithms (which are the exponents) are added.

2. The Division Law

The logarithm of a quotient is the difference between the logarithm of the numerator and the denominator: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

3. The Power Law

The logarithm of a number raised to a power is the power multiplied by the logarithm of the number: $\log_a x^k = k \log_a x$

Solving Equations with Logarithms

One of the most powerful applications of logarithms is solving equations where the variable is in the exponent.

Basic Exponential Equations

To solve an equation like $3^x = 20$, you can "take logs" of both sides:

1. Apply $\log_{10}$ to both sides: $\log_{10}(3^x) = \log_{10} 20$
2. Use the Power Law to bring $x$ down: $x \log_{10} 3 = \log_{10} 20$
3. Isolate $x$ by dividing: $x = \frac{\log_{10} 20}{\log_{10} 3}$
4. Calculate: $x \approx 2.73$.

Exam Tip: When using a calculator to solve these, always use the full decimal values provided by the calculator rather than rounded numbers to avoid significant rounding errors in your final answer.

Advanced Techniques: Substitution and Changing Base

For more complex equations, such as those that resemble quadratic equations (e.g., $7(5^x)^2 - 30(5^x) + 10 = 0$), you can use substitution. By letting $y = 5^x$, the equation becomes a standard quadratic: $7y^2 - 30y + 10 = 0$.

If your calculator does not have a specific base button, you can use the Change of Base formula to calculate any logarithm using base 10: $\log_a x = \frac{\log_{10} x}{\log_{10} a}$ For example, to find $\log_8 11$, you would calculate $\frac{\log_{10} 11}{\log_{10} 8} \approx 1.15$.