Solve:
2^(5x-6) = 7
My work:
log^(5x-6) = log7
5x - 6(log2) = log7
5x = log7 + 6(log2)
x = (log7 + log2^6) / 5
And textbook answer:
(log7) / (log2)
What did I do wrong?
you are wrong and the textbook answer, the way you stated is, is wrong.
2^(5x-6) = 7
log[2^(5x-6)] = log7
(5x-6)log2 = log7
5x-6 = log7/log2
5x = log7/log2 + 6
x = (log7/log2 + 6)/5
Without using a calculator, find the value of the following logarithmic expression.
log7 1
It appears that there is a mistake in your calculation:
Starting from the equation 2^(5x-6) = 7, you correctly applied the logarithm to both sides of the equation to take the base 2 exponent down. However, the mistake occurs when you simplify the logarithmic expression on the right-hand side.
Instead of using the property of logarithms as you did, which is log(a * b) = log(a) + log(b), you incorrectly applied it as log(a + b) = log(a) + log(b). This led to the incorrect expression: 5x = log7 + 6(log2). The exponent "6" should not be outside the logarithm.
To correct this mistake, you should have used the property of logarithms correctly:
2^(5x-6) = 7
log(2^(5x-6)) = log(7)
(5x - 6) * log(2) = log(7)
Now, when you solve for x, you should divide both sides by log(2):
5x - 6 = log(7) / log(2)
5x = log(7) / log(2) + 6
x = (log(7) + 6 * log(2)) / 5
Note that the base of the logarithm is not specified in your calculation, so it is assumed to be base 10 or log10. However, when finding the value of x, it is customary to use the natural logarithm (base e) or ln. In that case, the corrected expression would be:
x = (ln(7) + 6 * ln(2)) / 5
The textbook answer you provided of (log7) / (log2) seems to be simplifying further based on the given equation and not considering the information in your work.