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Exponential and logarithmic equations

Undo an exponential with a logarithm, and a logarithm with an exponential.

When the unknown is in the exponent, take a logarithm of both sides; the exponent comes down as a multiplier. When the unknown is inside a log, rewrite in exponential form. Always check the domain: logarithms need positive arguments.

Радни пример: 2^x = 8

Algebra · Exponential-Equations

Solve 2^x = 8

2^{x} = 8

Корак по корак

  1. 2^{x} = 8

    Start from the equation as given.

  2. 2^{x} - 8 = 0

    Bring everything to one side.

  3. 2^{x} - 8 = 0

    Isolate the exponential, then take a logarithm of both sides.

  4. x = 3

    Solve for the variable.

Откриј одговор.
x = 3

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Exponential and logarithmic equations

  1. Start from the equation as given.
  2. Bring everything to one side.
  3. Isolate the exponential, then take a logarithm of both sides.
  4. Solve for the variable.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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