Question

Solve each equation. $$ \log _{4} 64=x $$

Short answer

x = 3

Step-by-step solution

Understand the logarithmic equation

The equation to solve is given by: \(\log _{4} 64=x\). This means we need to determine what power \(x\) you have to raise 4 to in order to get 64.

Convert the logarithmic equation to an exponential form

To solve for \(x\), convert the logarithmic equation to its exponential form. The equation \(\log_{4} 64 = x\) in exponential form is: \[4^x = 64\]

Express 64 as a power of 4

Next, express 64 using a base of 4. Notice that 64 is equal to \(4^3\). So, \[64 = 4^3\]

Set the exponents equal to each other

Since both sides of the equation have the same base, set their exponents equal: \[4^x = 4^3\] implies \(x = 3\).

Key concepts

Exponential Form

To tackle logarithmic equations efficiently, knowing how to convert them into exponential form is essential. This is because converting it often simplifies the problem.
In the example \(\text{log}_{4} 64 = x\), what we are really asking is what power we need to raise 4 to in order to obtain 64.
To convert it to exponential form, you rearrange the equation: \[\text{log}_{b} (y) = x \rightarrow b^x = y\], where the base (\(b\)) raised to the power (\(x\)) equals the number (\(y\)).
For the equation \(\text{log}_{4} 64 = x\), converting it to exponential form gives: \({4^x} = 64\). This makes the equation more straightforward to solve.

Base Conversion

Understanding bases is another core part of solving logarithmic and exponential equations. Sometimes, converting a number to another base helps in simplifying computations.
For instance, in the problem \(\text{log}_{4} 64 = x\), we need to express 64 in terms of the base 4.
We do this by finding a power of 4 that equals 64. Notice that: \({4^3} = 64\).
Thus, the expression can be rewritten as: \(64 = 4^3\).
By converting 64 into the same base, we simplify the equation and make solving for the exponent easier.

Solving Equations

The final step in solving logarithmic equations involves setting the exponents equal, especially when the bases are the same.
After converting \(\text{log}_{4} 64 = x\) to \({4^x} = 64\) and expressing 64 as \(4^3\), we can compare the exponents.
Since both sides have the base 4, we equate the exponents: \({4^x} = 4^3\).
This implies that \(x = 3\).
The simplicity comes from the fact that equal bases yield equal exponents. This straightforward comparison is a crucial technique in solving these types of equations.