Logarithms and Their Properties

📐 Logarithms and Their Properties

Part 1 of 5 — What Is a Logarithm?

Topics in This Part

🔑 Key Concept: A logarithm answers a single question — "What exponent do I need?" The expression $\log_b x$ asks: to what power must I raise the base $b$ to get $x$?

The Definition

A logarithm is the inverse of an exponential. They say the exact same thing in two different forms:

$\log_b x = y \quad\Longleftrightarrow\quad b^y = x$

Read $\log_b x$ as "log base $b$ of $x$." The answer ($y$) is the exponent.

Examples

🔑 Key Idea: The base of the log becomes the base of the power, and the value of the log is the exponent. If you can rewrite a log as an exponent, you can evaluate it.

⚠️ The base $b$ must be positive and not equal to 1, and you can only take the log of a positive number. There is no real value for $\log_2(-4)$ or $\log_2 0$.

Concept Check 🎯

Evaluate the Log 🧮

Ask yourself: "the base to what power gives the inside?"

1) $\log_2 16 = \,?$ 2) $\log_5 125 = \,?$ 3) $\log_{10} 1000 = \,?$

Two Values to Memorize

Every base gives the same answer to these two, so learn them once:

$\log_b 1 = 0 \qquad\text{(because } b^0 = 1\text{)}$

$\log_b b = 1 \qquad\text{(because } b^1 = b\text{)}$

For example, $\log_7 1 = 0$ and $\log_7 7 = 1$ — no matter the base.

💡 A handy intuition: $\log_b x$ tells you roughly how many times you multiply $b$ by itself to reach $x$. You now have the one idea the whole topic rests on: a log is an exponent. In Part 2 we use the two famous bases that appear everywhere.

Match the Value 🔽

Use $\log_b 1 = 0$, $\log_b b = 1$, and the definition.

📐 Logarithms and Their Properties

Part 2 of 5 — Common Logs, Natural Logs & Inverses

🔑 The Idea: Two bases are so common they get their own shorthand. And because logs and exponentials are inverses, they undo each other — a fact that powers nearly every log problem you'll solve.

The Two Famous Bases

When you see $\log x$ with no base written, it means base 10. When you see $\ln x$, the base is the special number $e \approx 2.71828$.

Examples

$\log 1000 = \log_{10} 1000 = 3 \qquad (10^3 = 1000)$

$\ln e = \log_e e = 1 \qquad (e^1 = e)$

$\ln 1 = 0 \qquad (e^0 = 1)$

💡 Your calculator has two log buttons: LOG (base 10) and LN (base $e$). Both follow every rule in this lesson.

Concept Check 🎯

Logs and Exponentials Undo Each Other

Because $\log_b x$ and $b^x$ are inverse functions, applying one then the other returns you to the start. These are the inverse (cancellation) properties:

$b^{\,\log_b x} = x \qquad\text{and}\qquad \log_b\!\left(b^{\,x}\right) = x$

Examples

⚠️ The base must match. $\log_2(3^5)$ does not simplify to $5$, because the log base ($2$) and the power base ($3$) differ.

Cancel Them Out 🧮

Use $b^{\log_b x} = x$ and $\log_b(b^x) = x$ — but only when the bases match.

1) $\log_3\!\left(3^8\right) = \,?$ 2) $\ln\!\left(e^{6}\right) = \,?$ 3) $10^{\,\log 12} = \,?$

Will They Cancel? 🔽

Pick the simplified value. Watch whether the bases match.

📐 Logarithms and Their Properties

Part 3 of 5 — The Three Big Properties

🔑 The Engine of the Topic: Three rules let you break apart, combine, and move logs. They come straight from the laws of exponents, because a log is an exponent.

Product, Quotient, and Power Rules

For any valid base $b$ and positive numbers $M, N$:

Where they come from

Exponents add when you multiply: $b^x \cdot b^y = b^{x+y}$. Since logs are exponents, a product inside a log becomes a sum of logs. The other two rules follow the same way.

🔑 Key Idea: These rules turn multiplication into addition, division into subtraction, and powers into multiplication — they make hard logs simple.

Worked Example: Expanding

"Expand" means write a single log as a sum/difference of simpler logs.

Expand $\log_b\!\dfrac{x^3 y}{z}$

$\log_b\frac{x^3 y}{z} = \log_b(x^3 y) - \log_b z \quad\text{(quotient rule)}$

$= \log_b x^3 + \log_b y - \log_b z \quad\text{(product rule)}$

$= 3\log_b x + \log_b y - \log_b z \quad\text{(power rule)}$

⚠️ Watch the signs. Everything in the denominator gets subtracted. A common error is writing $+\log_b z$ instead of $-\log_b z$.

Concept Check 🎯

Apply One Rule 🧮

Use the property and a known value. Recall $\log_2 8 = 3$ and $\log_5 25 = 2$.

1) $\log_2\!\left(8^4\right) = 4\cdot\log_2 8 = \,?$ 2) $\log_5\!\left(25^3\right) = 3\cdot\log_5 25 = \,?$ 3) If $\log_b 6 = \log_b 2 + \log_b 3$ and $\log_b 2 = 0.43$, $\log_b 3 = 0.68$, then $\log_b 6 = \,?$

Worked Example: Condensing

"Condense" runs the rules backward — combine several logs into one.

Condense $3\log_b x + \log_b y - 2\log_b z$

First, use the power rule to move coefficients up as exponents:

$\log_b x^3 + \log_b y - \log_b z^2$

Then product rule joins the sums, quotient rule handles the subtraction:

$\log_b\frac{x^3 y}{z^2}$

💡 Strategy: When condensing, deal with coefficients first (power rule), then combine. Added terms go on top; subtracted terms go on the bottom.

Condense Step by Step 🔽

Combine $\log_b 5 + 2\log_b x - \log_b y$ into a single log.

📐 Logarithms and Their Properties

Part 4 of 5 — Change of Base & Solving Equations

🔑 Big Payoff: The change-of-base formula lets your calculator evaluate any log, and the properties let you solve equations where the unknown is trapped in an exponent.

The Change-of-Base Formula

Most calculators only do base 10 ($\log$) and base $e$ ($\ln$). To evaluate any other base, rewrite it:

$\log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}$

You can use either base on the right (10 or $e$) — the ratio comes out the same.

Example: $\log_2 50$

$\log_2 50 = \frac{\log 50}{\log 2} = \frac{1.699}{0.301} \approx 5.644$

Check: $2^5 = 32$ and $2^6 = 64$, so $\log_2 50$ should fall between $5$ and $6$. ✓

💡 The same numerator base goes on top and the same on bottom — never mix $\log$ on top with $\ln$ on bottom.

Concept Check 🎯

Solving Exponential Equations

When the variable is in the exponent, take the log of both sides, then use the power rule to bring it down.

Example: $3^x = 20$

$\log 3^x = \log 20 \;\Rightarrow\; x\log 3 = \log 20$

$x = \frac{\log 20}{\log 3} = \frac{1.301}{0.477} \approx 2.727$

Check: $3^{2.727}\approx 20$ ✓ (and it's sensibly between $3^2=9$ and $3^3=27$).

Solving Logarithmic Equations: $\log_2(x) = 5$

Rewrite in exponential form: $x = 2^5 = 32$.

⚠️ Always check log-equation answers in the original equation — the inside of a log must stay positive, so reject any solution that makes an argument $\le 0$.

Solve It 🧮

1) $\log_4 x = 3 \;\Rightarrow\; x = \,?$ (rewrite as a power) 2) $\log_3(x) = 0 \;\Rightarrow\; x = \,?$ 3) $10^x = 1000 \;\Rightarrow\; x = \,?$

Order the Steps 🔽

You're solving $5^x = 40$. Choose what happens at each stage.

📐 Logarithms and Their Properties

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read a log as an exponent, (2) use the common/natural logs and inverse properties, (3) expand and condense with the three rules, and (4) change base and solve equations. Let's put it together.

Quick Reference

⚠️ Remember: there is no rule that simplifies $\log_b(M+N)$ — the product rule applies to multiplication inside, never to a sum. $\log_b(M+N) \ne \log_b M + \log_b N$.

Spot the Mistake 🎯

Mixed Practice 🧮

1) $\log_2 64 = \,?$ 2) $\ln\!\left(e^{10}\right) = \,?$ 3) Solve $\log_5 x = 2 \;\Rightarrow\; x = \,?$

Build the Single Log 🔽

Condense $\log_b a + 3\log_b c - \log_b d$ one step at a time.

Exit Quiz ✅

Answer all three to finish the lesson.