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Mathematics › 🌌 Grade 11 › Logarithmic Slide Rule: Log Properties and Computations
🌌 Grade 11 · Lesson 5 of 12

Logarithmic Slide Rule: Log Properties and Computations

A logarithm asks “what exponent?” — and its magic is that it converts multiplication into addition, which is exactly why slide rules once did the work of calculators.

Grade 11Algebra 2 / Pre-Calculus
Logarithmic Slide Rule: Log Properties and Computations — illustration
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The big idea: A logarithm is the inverse of an exponential: log_b(x) answers “to what power must I raise b to get x?” Because exponents add when you multiply powers, logarithms turn products into sums, quotients into differences, and powers into multiplication. That single translation — from the hard world of multiplying to the easy world of adding — is what makes logarithms one of the most useful tools in mathematics.
🎯 By the end, you'll be able to
  • Convert between exponential form and logarithmic form
  • Apply the product, quotient, and power laws of logarithms
  • Use the change-of-base formula to evaluate logarithms with any base
  • Solve simple exponential and logarithmic equations using log properties
📎 You should already know
  • Exponents and laws of exponents
  • Inverse functions

A logarithm asks a question about exponents

Exponential notation by = x starts from the exponent and produces a value. A logarithm runs that backwards: given the value, it asks for the exponent. That is why the logarithm is the inverse of the exponential function.

Read logb(x) as: “the power to which the base b must be raised to give x.” So log2(8) = 3 because 23 = 8.

\[ \log_b(x) = y \iff b^{y} = x \]
The two forms say the same thing: the log is the exponent y that turns base b into x.
🔑 The defining move: log undoes exponent
Because the log and the exponential are inverses, they cancel: logb(bx) = x and blogb(x) = x. This is the lever you pull to get a variable out of an exponent when solving equations.

Turning multiplication into addition

Here is the property that changed computation forever. When you multiply two powers of the same base you add their exponents: bm · bn = bm+n. Translate that through the logarithm and multiplication on the inside becomes addition on the outside.

The old slide rule was built on exactly this: it laid numbers on a logarithmic scale so that sliding one length past another physically added the logs — and thereby multiplied the numbers.

🎮 Logarithmic Slide Rule LIVE
A log turns multiplication into addition — slide along the log scale to see why.
\[ \log_b(MN) = \log_b M + \log_b N \]
The product law: the log of a product is the sum of the logs. This is the heart of the slide rule.
✨ The three laws of logarithms
Product: logb(MN) = logbM + logbN. Quotient: logb(M/N) = logbM − logbN. Power: logb(Mp) = p·logbM. Each mirrors an exponent law: multiply→add, divide→subtract, raise-to-a-power→multiply.

Change of base

Calculators usually only offer base 10 (log) and base e (ln). To evaluate a log in any other base, rewrite it as a ratio of logs you can compute. Any convenient base works for the top and bottom, as long as they match.

\[ \log_b(x) = \dfrac{\log_c(x)}{\log_c(b)} \]
Change-of-base: to compute log base b, divide the log of x by the log of b in any base c you like.
📝 Worked example: Write log₂(5) + log₂(3) − log₂(6) as a single logarithm and evaluate it.
  1. Combine the sum with the product law: \( \log_2 5 + \log_2 3 = \log_2(5 \cdot 3) = \log_2 15 \).
  2. Combine the difference with the quotient law: \( \log_2 15 - \log_2 6 = \log_2\!\left(\dfrac{15}{6}\right) = \log_2 2.5 \).
  3. So the expression equals \( \log_2 2.5 \), which is the exponent putting 2 up to 2.5 — a value between 1 and 2 (since \(2^1=2\) and \(2^2=4\)).
✓ It simplifies to <strong>log<sub>2</sub>(2.5) &asymp; 1.32</strong>.
📝 Worked example: Solve 3^x = 20 for x.
  1. Take a logarithm of both sides to bring x down from the exponent: \( \ln(3^x) = \ln 20 \).
  2. Apply the power law on the left: \( x \ln 3 = \ln 20 \).
  3. Divide to isolate x: \( x = \dfrac{\ln 20}{\ln 3} \).
✓ <strong>x = ln 20 / ln 3 &asymp; 2.73</strong> — a check: 3<sup>2.73</sup> is about 20.
⚠️ There is no law for the log of a sum
The product law is about the log of a product. It is not true that log(M + N) equals log M + log N. The log of a sum cannot be split at all — leave log(M + N) exactly as it is. Also, you can only take logs of positive numbers.

Check your understanding

1. What is log₃(81)?
log₃(81) asks 3 to what power is 81; since 3⁴ = 81, the answer is 4.
2. Which is equal to log(M) + log(N)?
The product law: a sum of logs is the log of the product, log(MN).
3. Rewrite log(x⁵) using a log law.
The power law moves the exponent out front: log(x⁵) = 5·log(x).
4. To solve 2^x = 7, a good first step is to…
Taking a log of both sides lets the power law bring x down out of the exponent.
5. Using change of base, log₅(12) equals…
Change of base gives log₅(12) = log(12) / log(5), with any consistent base on top and bottom.
✅ Key takeaways
  • A logarithm log_b(x) is the exponent that turns base b into x — the inverse of the exponential.
  • Log and exponential cancel: log_b(bˣ) = x, which pulls a variable out of an exponent.
  • Product law: log(MN) = log M + log N; quotient law subtracts; power law multiplies by the exponent.
  • Change of base: log_b(x) = log(x)/log(b) in any convenient base.
  • There is no law for the log of a sum, and logs are only defined for positive numbers.