Number and Algebra - Exponents and Logarithms

Laws of Indices: These rules define how to manipulate expressions with exponents. When multiplying terms with the same base, add the exponents ($a^m \times a^n = a^{m+n}$); when dividing, subtract them ($\frac{a^m}{a^n} = a^{m-n}$). Raising a power to another power involves multiplying the exponents ($(a^m)^n = a^{mn}$). Visually, an exponent represents repeated multiplication, and these laws simplify the counting of those repetitions.

Rational and Negative Exponents: A negative exponent represents the reciprocal of the base raised to the positive power ($a^{-n} = \frac{1}{a^n}$). Fractional exponents represent roots: the denominator indicates the root degree and the numerator indicates the power ($a^{\frac{m}{n}} = \sqrt[n]{a^m}$). Visually, $a^{\frac{1}{2}}$ represents the side length of a square with area $a$.

Logarithm Definition and Inverse Relationship: A logarithm is the inverse of an exponential function. The statement $\log_a(x) = b$ is equivalent to $a^b = x$. Visually, if you look at the graph of $y = a^x$, its reflection over the diagonal line $y = x$ produces the graph of $y = \log_a(x)$. This means the domain of the exponential (all real numbers) becomes the range of the logarithm, and the range of the exponential ($y > 0$) becomes the domain of the logarithm ($x > 0$).

Properties of Logarithms: Logarithms follow specific rules derived from index laws: the Product Rule ($\log_a(xy) = \log_a x + \log_a y$), the Quotient Rule ($\log_a(\frac{x}{y}) = \log_a x - \log_a y$), and the Power Rule ($\log_a(x^k) = k \log_a x$). These properties allow complex multiplicative relationships to be converted into simpler additive ones.

Natural Logarithms and Euler's Number: The constant $e \approx 2.71828$ is a transcendental number that serves as the base for the natural logarithm, denoted as $\ln(x) = \log_e(x)$. In calculus and growth modeling, $e^x$ is unique because its rate of change (slope) at any point is equal to the value of the function itself. Visually, the graph of $y = e^x$ passes through $(0, 1)$ with a gradient of $1$.

Solving Exponential Equations: When variables are in the exponent, such as $a^x = b$, we apply logarithms to both sides to 'bring down' the exponent using the power rule. If the bases can be made equal, we can simply equate the exponents. Visually, solving $a^x = b$ is finding the $x$-coordinate where the exponential curve $y = a^x$ intersects the horizontal line $y = b$.

Logarithmic Graphs and Asymptotes: The graph of $f(x) = \log_a(x)$ always passes through the point $(1, 0)$ because any base raised to the power $0$ is $1$. It has a vertical asymptote at $x = 0$ (the $y$-axis), meaning the function is undefined for non-positive values of $x$. As $x$ increases, the graph increases but the slope becomes flatter and flatter, representing very slow growth.

Change of Base Formula: To calculate or simplify logarithms with bases not available on a standard calculator, the change of base formula is used: $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$. This allows any logarithm to be expressed in terms of common logarithms (base $10$) or natural logarithms (base $e$).