Logarithms

Understanding Logarithms

Logarithms give us a way to “undo” exponential functions and to work with very large or very small numbers more conveniently. In this chapter, the focus is on what logarithms are, how they are written, and the basic rules they satisfy. Later subsections will use these ideas to develop laws of logarithms and methods for solving equations involving logs and exponentials.

The idea of a logarithm

Suppose you know an exponential relationship
$$
b^x = N
$$
where $b$ is a positive number not equal to $1$, $x$ is an exponent, and $N$ is the result.

A logarithm answers the question:

“What exponent $x$ makes $b^x$ equal to $N$?”

We write this as:
$$
\log_b N = x
$$
and read it as “log base $b$ of $N$ equals $x$.”

This definition is equivalent to:
$$
\log_b N = x \quad \text{if and only if} \quad b^x = N.
$$

So:

• The base of the logarithm is $b$.
• The argument (or input) of the logarithm is $N$.
• The value of the logarithm is the exponent $x$.

Example interpretations

• $\log_2 8 = 3$ because $2^3 = 8$.
• $\log_{10} 1000 = 3$ because $10^3 = 1000$.
• $\log_3 1 = 0$ because $3^0 = 1$.
• $\log_5 \left(\dfrac{1}{25}\right) = -2$ because $5^{-2} = \dfrac{1}{25}$.

The logarithm does not “create” a new relationship; it restates an exponential relationship in a different form.

Logarithmic and exponential forms

You should be able to move back and forth between exponential form and logarithmic form.

• Exponential form: $b^x = N$
• Logarithmic form: $\log_b N = x$

These are two ways of saying the same thing.

Converting between forms

1. From logarithmic to exponential:

If you see $\log_b N = x$, rewrite as $b^x = N$.

• Example: $\log_4 16 = 2$ means $4^2 = 16$.
• Example: $\log_7 \dfrac{1}{49} = -2$ means $7^{-2} = \dfrac{1}{49}$.

2. From exponential to logarithmic:

If you see $b^x = N$, rewrite as $\log_b N = x$.

• Example: $3^4 = 81$ can be written $\log_3 81 = 4$.
• Example: $10^{-2} = 0.01$ can be written $\log_{10} 0.01 = -2$.

Being fluent in switching forms is essential for later solving equations with logs and exponentials.

Domain and base conditions for logarithms

Logarithms are not defined for all real numbers. The basic restrictions for $\log_b N$ (with real numbers) are:

• The base $b$ must be positive and not equal to $1$:
  $$
  b > 0, \quad b \neq 1.
  $$
  Bases like $, $, and $e$ (a special constant approximately .71828$) are typical.
• The argument $N$ must be positive:
  $$
  N > 0.
  $$
  We do not take the logarithm of zero or a negative number in the real-number setting.

So expressions like $\log_2(-4)$ or $\log_5(0)$ are undefined in real numbers.

Special bases: common and natural logarithms

Although $\log_b$ can use any valid base $b$, two bases are especially important.

Common logarithm (base 10)

The common logarithm uses base $10$:
$$
\log_{10} x.
$$

It is usually written simply as:
$$
\log x
$$
without writing the base. Unless otherwise stated, $\log x$ means base $10$ in most algebra contexts.

Examples:

• $\log 100 = 2$ because $10^2 = 100$.
• $\log 0.01 = -2$ because $10^{-2} = 0.01$.

Base $10$ is convenient for scientific notation and decimal-based calculations.

Natural logarithm (base $e$)

The natural logarithm uses base $e$:
$$
\log_e x.
$$

It is written as:
$$
\ln x
$$
(read “ell-en x”).

Examples:

• $\ln 1 = 0$ because $e^0 = 1$.
• If $e^2 \approx 7.389$, then $\ln 7.389 \approx 2$.

The base $e$ and the natural logarithm $\ln$ become especially important in calculus and in models involving continuous growth or decay.

Basic values and patterns

Using the definition of logarithms, we can identify some key values and general patterns.

Logs of 1

For any valid base $b$,
$$
\log_b 1 = 0
$$
because $b^0 = 1$.

So:

• $\log 1 = 0$,
• $\ln 1 = 0$,
• $\log_2 1 = 0$, and so on.

Logs of the base

For any valid base $b$,
$$
\log_b b = 1
$$
because $b^1 = b$.

So:

• $\log 10 = 1$,
• $\ln e = 1$,
• $\log_3 3 = 1$.

Logs of powers of the base (integer exponents)

If $N$ is an integer power of the base, equations are easy to evaluate mentally:
$$
\log_b (b^k) = k
$$
for any real number $k$.

Examples:

• $\log_2 32 = \log_2 (2^5) = 5$.
• $\log_3 \dfrac{1}{27} = \log_3 (3^{-3}) = -3$.
• $\log 100000 = \log (10^5) = 5$.

You are reading off “what exponent of $b$ gives this number?”

Estimating logarithms

Not all logarithms are “nice” integers. For example, $\log_{10} 3$ is not an integer. Without a calculator, you can estimate by comparing to nearby powers of the base.

Example with base $10$:

• $10^0 = 1$, $10^1 = 10$.
• $3$ is between $1$ and $10$.
• So $0 < \log 3 < 1$.

Example with base $2$:

• $2^2 = 4$, $2^3 = 8$.
• $5$ is between $4$ and $8$.
• So $2 < \log_2 5 < 3$.

This way of thinking (“between which powers of the base does this number lie?”) is useful conceptually even when you will normally use a calculator for actual numerical values.

Interpretation in terms of size and scaling

Logarithms often appear when we:

• count how many times something must be multiplied by a base to reach a certain size,
• convert multiplicative growth into additive steps.

A simple interpretation:

• If $\log_2 N = 10$, then $N$ is obtained by doubling (multiplying by $2$) ten times starting from $1$:
  $$
  N = 2^{10}.
  $$
• If $\log_{10} N = -3$, then $N$ is $10^{-3}$, or $0.001$; it lies three decimal places to the right of $1$ when written as a small decimal.

Many “log scales” used in real applications (such as some measures in science and engineering) take advantage of this idea of counting multiplicative factors by using logarithms.

Graphs of basic logarithmic functions

You will later study functions in more detail, but for now it is useful to know the general shape of the basic logarithmic function.

Consider $y = \log_b x$ with $b > 1$ (e.g., $b=2$ or $b=10$):

• The domain is $x > 0$ (the graph is only to the right of the $y$-axis).
• The graph passes through $(1, 0)$ because $\log_b 1 = 0$.
• The graph increases slowly: as $x$ gets larger, $y$ increases but at a decreasing rate.
• There is a vertical asymptote at $x = 0$; the graph gets very low (goes to $-\infty$) as $x$ approaches $0$ from the right.

For $0 < b < 1$, the logarithmic graph is reflected horizontally and is decreasing instead of increasing, but in algebra courses the most common bases used as functions are $b>1$.

Relationship with exponential functions

Logarithmic and exponential functions with the same base are inverse functions of each other.

• If $y = b^x$, then $x = \log_b y$.
• More compactly:
  $$
  \log_b(b^x) = x \quad \text{and} \quad b^{\log_b x} = x
  $$
  for all $x$ in the appropriate domain.

This inverse relationship is the foundation for later work:

• undoing exponentials using logarithms,
• rewriting exponential equations in logarithmic form to solve for exponents,
• converting between $b^x$ and $\log_b x$ when analyzing graphs.

Understanding that a logarithm “undoes” an exponential with the same base is the central idea that ties this entire chapter together.

Summary of key points

• $\log_b N$ is the exponent you put on $b$ to get $N$: $\log_b N = x$ means $b^x = N$.
• Valid bases satisfy $b > 0$ and $b \neq 1$, and the argument $N$ must be positive.
• Converting between exponential and logarithmic forms is essential: $b^x = N \iff \log_b N = x$.
• Common log: $\log x = \log_{10} x$; natural log: $\ln x = \log_e x$.
• Important values: $\log_b 1 = 0$, $\log_b b = 1$, and $\log_b (b^k) = k$.
• Logarithms and exponentials with the same base are inverse to each other.

Later sections on logarithmic laws and solving logarithmic equations will build directly on these ideas.