U35 Exponential and Logarithmic Functions: Essential Formulas

Exponential Function

For $a > 0$ and $a \neq 1$, the exponential function with base $a$ is defined as $y = a^x$, where $x$ is any real number. The graph passes through $(0, 1)$ and has a horizontal asymptote at $y=0$.

Logarithmic Function

The logarithmic function is the inverse of the exponential function. If $y = a^x$, then $x = \log_a y$, for $a > 0$, $a \neq 1$, and $y > 0$. The graph passes through $(1, 0)$ and has a vertical asymptote at $x=0$.

Fundamental Laws

For $a > 0$, $a \neq 1$, $M > 0$, $N > 0$, and any real number $p$, the following laws hold. These are essential for simplifying and solving logarithmic equations.

Additionally, the change of base formula is $\log_a b = \frac{\log_c b}{\log_c a}$ for any valid base $c$.

The Constant $e$

The number $e \approx 2.71828$ is the base of the natural logarithm. The function $y = e^x$ is its own derivative, making it central to calculus and modeling continuous growth/decay.

Natural Logarithm $\ln x$

The natural logarithm is the logarithm to the base $e$, denoted as $\ln x = \log_e x$. It shares all properties of general logarithms.

Key Techniques

To solve equations involving exponentials and logarithms, common strategies include: expressing both sides with the same base, taking logarithms on both sides, and using substitution (e.g., let $y = a^x$). Always check the domain: the argument of any logarithm must be positive.

Example: To solve $2^{x+1} = 5$, take $\log$ on both sides: $(x+1)\log 2 = \log 5$, so $x = \frac{\log 5}{\log 2} - 1$.