Logarithm

Accueil » Logarithm
Logarithm: meaning, method and solved example (Algebra 2) – Maths PDF

In short: A logarithm is the exponent that a chosen base must carry to produce a given number: writing log base b of x equals y says exactly the same thing as b to the power y equals x.

Algebra 2Functions and GraphsF-LE.A.4

What is Logarithm?

A logarithm answers the question “what exponent?” If you know that 2 multiplied by itself five times is 32, then the logarithm of 32 in base 2 is 5. It reads an exponential statement backward.

For a base \(b > 0\) with \(b \neq 1\) and a number \(x > 0\), \(\log_b x = y\) means \(b^y = x\). The input must be positive, and the result may be negative or fractional. Logs in base 10 are written without a base, and the natural kind uses base \(e\).

How to spot it

  • The unknown sits in an exponent, as in 3^x = 200, and a plain root cannot reach it.
  • A notation like log_5 125 appears and you must supply a single number.
  • A scale grows by multiples instead of sums, as with earthquake magnitude or sound level in decibels.
  • A question asks you to rewrite an exponential statement in another form.

The method

  1. Name the base, the number inside, and the answer you want: \(\log_b x = ?\).
  2. Translate to the exponential form \(b^? = x\).
  3. Rewrite \(x\) as a power of \(b\), or test small exponents, until the power matches.
  4. State the exponent as the value, then confirm by raising the base to it.
exponential form25 = 32logarithmic formlog2 32 = 5the logarithm is the exponent: 5
Both statements say the same thing: base 2, exponent 5, result 32.

Logarithm: solved example

Problem

Evaluate \(\log_6 216\) and \(\log_{10} 0.001\), then solve \(\log_5 x = -2\).

Solution

Since \(6^3 = 216\), \(\log_6 216 = 3\). Since \(0.001 = 10^{-3}\), \(\log_{10} 0.001 = -3\).

For the equation, rewrite \(\log_5 x = -2\) as \(x = 5^{-2} = \frac{1}{25}\). The three answers are 3, \(-3\) and \(\frac{1}{25}\).

Try it

Question 1

Find \(\log_3 81\).

Show the answer

\(3^4 = 81\), so the value is 4.

Question 2

Write \(2^9 = 512\) in logarithmic form.

Show the answer

\(\log_2 512 = 9\).

Logarithm vs. Exponential Function

An exponential function such as \(2^x\) takes an exponent and returns a number. The logarithm runs the other way: it takes the number and returns the exponent. Their graphs are mirror images across the line \(y = x\), which is why each one undoes the other.

Exponential Function

Keep going

Lessons

Worksheets and quizzes

Related words

Whole dictionary: A–Z · Algebra 2 · Functions and Graphs

Voter.. page