What is the logarithmic form of the equation 4^x = 64 in base 10?

To convert the exponential equation 4^x = 64 into its logarithmic form in base 10, we first need to understand the relationship between exponential and logarithmic expressions.

In general, the exponential equation a^b = c can be rewritten in logarithmic form as log_a(c) = b. Here, a is the base, b is the exponent, and c is the result of the exponential equation.

In our case, we can express 64 as a power of 4. Notice that:

64 = 4³

Thus, we can rewrite the original equation as:

4^x = 4³

Since the bases are the same, we can set the exponents equal to each other:

x = 3

Now, to express this in logarithmic form, we can derive the equivalent form. Since we established that 4^x = 64, we can say:

log₄(64) = x

However, since the question specifically requests the logarithmic form in base 10, we need to use the change of base formula, which states:

log_a(b) = log_c(b) / log_c(a)

Applying this formula with a = 4, b = 64, and c = 10, we can calculate:
log₁₀(64) / log₁₀(4) = x

So, in conclusion, the logarithmic form of the equation 4^x = 64 in base 10 can be expressed as:

x = log₁₀(64) / log₁₀(4)

This gives you an equivalent way to represent the relationship established by the original exponential equation.