Express log to the base three-sevenths of 27 over 343 equals three in its equivalent exponential form.
We recall that in order to convert an expression from logarithmic form to exponential form, we use the following rule. If log base 𝑎 of 𝑥 is equal to 𝑏, then 𝑥 is equal to 𝑎 to the power of 𝑏. In this question, the base 𝑎 is equal to three-sevenths. Our value of 𝑏 is three. The value of 𝑥 is 27 over 343. This means that the equivalent exponential form is 27 over 343 is equal to three-sevenths cubed.
We could check this answer by cubing three-sevenths. We recall that to cube any fraction 𝑎 over 𝑏, we can cube the numerator and denominator separately. Three-sevenths cubed is equal to three cubed over seven cubed. We know that three cubed is 27. Seven cubed is 343. A seven multiplied by seven is 49, and multiplying this by seven gives us 343. This confirms that our exponential form is correct.
We can use this rule to convert any expression from logarithmic form to exponential form or vice versa.