1/3log(base7) x=2log(base7)4

Solve for x?

To solve the equation 1/3log(base7)x = 2log(base7)4, we need to use logarithmic properties to simplify the equation. Let's break it down step-by-step:

Step 1: Apply the power rule of logarithms
According to the power rule, log(basea)b^c = c * log(basea)b. We can use this rule to simplify the equation.

Applying the power rule to 2log(base7)4 gives us:
2log(base7)4 = log(base7)4^2
= log(base7)16

Step 2: Simplify the equation
Now we have the equation 1/3log(base7)x = log(base7)16. To further simplify, we can use the division rule of logarithms. According to the division rule, log(basea)b / log(basea)c = log(c)b.

Using the division rule on the left side of the equation gives us:
1/3log(base7)x = log(base7)16
(log(base7)x)^(1/3) = log(base7)16

Step 3: Convert to exponential form
To solve for x, we can rewrite the equation in exponential form. In exponential form, log(basea)b = c is equivalent to a^c = b.

Rewriting the equation in exponential form gives us:
(log(base7)x)^(1/3) = 16

Step 4: Solve for x
To solve for x, we need to remove the exponent on the left side of the equation. To do that, we raise both sides to the power of 3:

((log(base7)x)^(1/3))^3 = 16^3
log(base7)x = 16^3
x = 7^(16^3)

Therefore, the solution to the equation 1/3log(base7)x = 2log(base7)4 is x = 7^(16^3).