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Exponents are a way of expressing repeated multiplication of the same number. For example, the expression \( 3^2 \) means \( 3 \) multiplied by itself, which equals 9. In general, when you see a number like \( a^b \), the \( b \) is the exponent and it tells us how many times to multiply \( a \) by itself.
Exponents have several key properties that are good to understand:

• Multiplying Powers: \( a^m \times a^n = a^{m+n} \)
• Dividing Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)

Understanding exponents is essential when working with logarithms because a logarithm essentially seeks the exponent you would use to reach a given value.

The change of base formula is a useful technique in logarithms that allows us to calculate a logarithm with any base by converting it to a fraction of two logarithms with a new, common base. If you're dealing with logarithms that don't result in nice whole numbers or easily recognizable values, the change of base formula becomes very handy.
The formula is:

• \( \log_b a = \frac{\log_c a}{\log_c b} \)

Here, \( c \) is the new base you choose, often 10 (common logarithm) or \( e \) (natural logarithm) because these are readily available on calculators.
For example, we used it to solve \( \log_{16} 8 \) in the exercise. By expressing both 8 and 16 as powers of 2 (\( 8 = 2^3 \) and \( 16 = 2^4 \)), we were able to simplify the problem using the change of base formula. This process showed that the answer is \( \frac{3}{4} \).

Logarithmic functions are the inverse of exponential functions. This means that while exponential functions take a base number and an exponent to produce a result, logarithmic functions take a result and a base to find the exponent. If you know that \( b^n = a \), then \( \log_b a = n \).
Logarithmic functions have several important properties:

• Product Rule: \( \log_b(xy) = \log_b x + \log_b y \)
• Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• Power Rule: \( \log_b(x^a) = a \times \log_b x \)

These properties allow you to break down complex logarithmic expressions into simpler parts, which can make solving logarithmic equations easier. Logarithmic functions are a foundational concept in algebra, helping us understand scaling and growth processes, especially in science and engineering.