1. The Big Idea
This lesson is really about one central question:
"If I know a base and a result, what exponent got me there?"
That is what logarithms do.
Exponents and logarithms are inverse operations — they undo each other.
Exponential form
$$2^3 = 8$$
Logarithmic form
$$\log_2 8 = 3$$
These two statements mean the same thing — just written in different forms.
- • The exponential form tells you the result of repeated multiplication.
- • The logarithmic form tells you which exponent created that result.
2. What an Exponent Means
An exponent tells you how many times a base is multiplied by itself.
$$2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$$
• base: $2$
• exponent: $4$
• value: $16$
$$10^3 = 1000$$
$10$ is multiplied by itself $3$ times.
Negative exponents
A negative exponent means you take the reciprocal.
$$10^{-2} = \frac{1}{10^2} = \frac{1}{100}$$
3. What a Logarithm Means
A logarithm asks:
"What exponent do I put on this base to get that number?"
$$\log_2 8 = 3$$
Read as: "log base 2 of 8 equals 3"
Meaning: "2 to what power equals 8?"
Since $2^3 = 8$, the answer is $3$.
Whenever you see a logarithm, translate it into a question about an exponent.
General Pattern
$$\log_b a = c \quad \Longleftrightarrow \quad b^c = a$$
These two forms express exactly the same relationship.
4. Saying Logarithms in Words
When you see $\log_3 81$, you say: "log base 3 of 81."
What it means is: "3 to what power equals 81?"
Since $3^4 = 81$:
$$\log_3 81 = 4$$
When a problem asks you to explain a logarithm, give three things:
- • How to say it aloud
- • What exponent question it is asking
- • What the value is
5. Why Base 2 Shows Up So Much
Base 2 is useful because powers of 2 appear often in math, science, and technology. Knowing these by heart lets you estimate logarithms quickly.
| Expression | Value |
|---|---|
| $2^1$ | $2$ |
| $2^2$ | $4$ |
| $2^3$ | $8$ |
| $2^4$ | $16$ |
| $2^5$ | $32$ |
| $2^6$ | $64$ |
| $2^7$ | $128$ |
Estimate $\log_2 20$.
Ask: "2 to what power gives 20?"
$2^4 = 16 \quad$ and $\quad 2^5 = 32$
Since $20$ is between $16$ and $32$:
$$4 < \log_2 20 < 5$$
6. Estimating Exponential Expressions
When you see an exponent that is not a whole number — like $2^{4.5}$ — the value will fall between the two surrounding whole-number powers.
Estimate $2^{4.5}$.
$2^4 = 16 \quad$ and $\quad 2^5 = 32$
So $2^{4.5}$ is somewhere between $16$ and $32$.
Key insight
- • If the exponent increases, the value increases
- • If the exponent is between two numbers, the result is between the corresponding powers
Fractional exponents are not strange — they just describe values between familiar powers.
7. Missing Exponents
Many of these questions are really just asking you to find the missing exponent. Even if a problem does not use log notation, treat it like a logarithm question.
Solve: $10^{\,?} = 100$
"10 to what power equals 100?"
Since $10^2 = 100$, the missing exponent is $\mathbf{2}$.
Solve: $2^{\,?} = \dfrac{1}{2}$
"2 to what power equals one-half?"
Since $2^{-1} = \dfrac{1}{2}$, the missing exponent is $\mathbf{-1}$.
When you see a blank exponent, think of it exactly like a logarithm question.
8. Writing Numbers as Powers
Recognizing numbers as powers of a base makes logarithms much easier to evaluate.
Powers of 10
Powers of 2
Evaluate $\log_2 16$.
Recognize that $16 = 2^4$.
$$\log_2 16 = 4$$
9. Exponential Growth
An investment or population that grows by the same percent each period is modeled exponentially.
Growth Model
$$A = P(1+r)^t$$
$P$ = initial amount
$r$ = growth rate (as a decimal)
$t$ = time
$A$ = final amount
Something grows at $5\%$ per year. After $t$ years:
$$A = P(1.05)^t$$
Because $1 + r = 1 + 0.05 = 1.05$.
Finding the rate
If an investment multiplies by a factor over time, set up the equation and solve for $r$. For example, "increase by a factor of $5$ in $20$ years" means:
$$(1+r)^{20} = 5$$
This is where logarithms help you solve for the unknown exponent.
10. Exponential Decay
Decay means the amount decreases by the same percent over equal time intervals.
Decay Model
$$A = P(1-r)^t$$
Same as the growth model, but you subtract the rate instead of adding it.
Common examples of decay:
Something loses $20\%$ each period. The multiplier is:
$$1 - r = 1 - 0.20 = 0.8$$
So the model is $A = P(0.8)^t$. Each period, only $80\%$ remains.
11. Half-Life
Half-life is the time it takes for the amount to become half of its current amount.
Starting with $800$ mg, the substance halves each period:
Each arrow represents one half-life. The amount is multiplied by ½ each time.
Exponential vs. linear decay
Exponential (halving)
800 → 400 → 200 → 100
Equal factors each time
Linear (subtracting)
800 → 700 → 600 → 500
Equal amounts subtracted
12. How to Read a Decay Graph
When looking at a decay graph, do not look for equal drops in amount. Look for equal factors.
How to find the half-life from a graph
- Pick any point on the graph and note its amount
- Find where the amount is exactly half of that
- Measure how much time passed — that is the half-life
If a graph starts at $800$ mg and reaches $400$ mg after about $1.5$ weeks, then the half-life is about $1.5$ weeks.
You would then expect $200$ mg to appear about $1.5$ weeks after that.
Reading the graph is really about matching amounts that are halves of each other, then measuring the time between them.
13. Why Logarithms Are Useful in Growth and Decay
Logs are useful when the exponent is unknown.
How long does it take for something growing at $8\%$ to multiply by $5$?
$$(1.08)^t = 5$$
The exponent $t$ is unknown. Logarithms let you isolate and solve for it.
Logs solve for time, rate, or exponent in exponential situations.
That is one of the main reasons logarithms matter in real life.
14. Important Core Facts
$\log_b 1 = 0$
Because $b^0 = 1$. The exponent needed to get $1$ is always $0$.
$\log_b b = 1$
Because $b^1 = b$. The exponent needed to get the base itself is always $1$.
Inverse relationship
$$b^{\log_b x} = x \qquad \text{and} \qquad \log_b(b^x) = x$$
Exponents and logarithms undo each other.
15. Relationship Between Exponential and Logarithmic Forms
This is one of the main ideas in the lesson. The two forms carry identical meaning — only the notation changes.
Exponential form
$$2^6 = 64$$
Base $2$, exponent $6$, gives $64$
Logarithmic form
$$\log_2 64 = 6$$
Which exponent on base $2$ gives $64$?
Nothing changed in meaning. Only the form changed.
- • Exponential form gives the result from a base and exponent
- • Logarithmic form asks which exponent produces the result
16. How to Think Through These Problems
Use these mental triggers to guide your thinking:
When you see an exponential expression…
"Multiply the base by itself this many times."
When you see a logarithm…
"What exponent gives me this number?"
When you see growth…
"Same percent increase over equal time intervals."
When you see decay…
"Same percent decrease over equal time intervals."
When you see half-life…
"How long until the amount is cut in half?"
17. Final Summary
Exponents tell you how many times a base is used as a factor.
Logarithms tell you what exponent is needed to produce a number.
Exponential growth happens when a quantity increases by the same percent each time period — model: $A = P(1+r)^t$.
Exponential decay happens when a quantity decreases by the same percent each time period — model: $A = P(1-r)^t$.
Half-life is the time required for a decaying quantity to become half its current amount.
Exponential and logarithmic equations are equivalent forms of the same relationship: $b^c = a \;\Longleftrightarrow\; \log_b a = c$.
Core facts: $\log_b 1 = 0$, $\;\log_b b = 1$, $\;b^{\log_b x} = x$, $\;\log_b(b^x) = x$.