Question

Solve each equation. $$ \log _{x} 16=2 $$

Short answer

x = 4

Step-by-step solution

- Understand the Equation

The given equation is \( \log _{x} 16=2 \). This notation states that the logarithm of 16 with base x is 2.

- Convert the Logarithmic Equation to Exponential Form

Recall that \( \log _{b} a = c \) can be rewritten in exponential form as \( b^{c} = a \). Thus, \( \log _{x} 16=2 \) can be written as \( x^{2} = 16 \).

- Solve for x

Solve the equation \( x^{2} = 16 \). Take the square root of both sides to obtain \( x = \pm 4 \). However, since the base of a logarithm cannot be negative or 1, the only valid solution is \( x = 4 \).

Key concepts

Understanding Logarithms

Logarithms are a fundamental concept in mathematics. They are the inverse operations of exponentiation. In simple terms, if you know the logarithm of a number, you can find the exponent that the base must be raised to, to get that number.
For example, in the equation \( \log _x 16=2 \), it asks: 'To what power must the base x be raised, to produce 16?'.
The logarithm tells us this power (in this case, 2). To further break it down:

• \( \log _b a = c \) is the same as \ b^c = a. \
• \( b\) is the base of the logarithm.
  In our example: \( x \).
  \( a \) is the result of the exponentiation.
  In our example: \( 16 \).
  \( c \) is the exponent.
  In our example: \( 2 \).

Understanding this notation and how to switch between logarithms and exponentiation is crucial.

Converting Logarithmic to Exponential Form

Converting a logarithmic equation to an exponential form can make it easier to solve.
For our example, we start with the logarithmic equation:
\( \log_{x} 16=2\).
Recall that logarithmic equations can be rewritten as exponential equations using the formula:
\( \log_{b} a = c\) becomes \( b^{c} = a\).
Applying this to our equation:
\( x^{2} = 16\).
This means that the base x, raised to the power 2, equals 16.

Using this method:
• You take the base (\( x \)).
  
• You take the exponent (\( 2 \)).
  
• Set it equal to the number (\(16 \)).

By converting to exponential form, you simplify the problem, making it easier to solve.

Solving Logarithmic Equations

Once we've converted the logarithmic equation to its exponential form, solving it involves traditional algebraic methods.
For \( x^2 = 16 \), we solve by taking the square root of both sides.
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