Suppose 2^x = 10.
Compute 8^x
i get 8 ^((ln 10) / (ln 2) ) but this was incorrect? i do think it was right, ideas or something im missing?
8 = 2^3
(2^3)^x = (2^x)^3 = 10^3
recall how to change base of logs.
8 ^((ln 10) / (ln 2) ) = 8^log210
= (2^3)^210
= 2^(210)^3
But, 2^210 = 10
so, it's still 10^3
2^x = 10.
x*Log2 = Log10,
X = Log10/Log2 = 3.32192809.
8^x = 8^(3.32192809) = 1000.
To compute 8^x when given the equation 2^x = 10, we can use the relation between the two bases.
Let's start by rewriting 8^x as (2^3)^x. Using the property of exponents, we know that (a^m)^n = a^(m * n). Therefore, (2^3)^x = 2^(3 * x).
Now, from the given equation 2^x = 10, take the logarithm of both sides to solve for x. We can choose any base for the logarithm, but using the natural logarithm (ln) is common:
ln(2^x) = ln(10).
Applying the exponent property of logarithms (ln(a^b) = b * ln(a)), we get:
x * ln(2) = ln(10).
Now, divide both sides by ln(2):
x = ln(10) / ln(2).
Finally, substitute this value of x into 2^(3 * x) to get the result:
8^x = 2^(3 * (ln(10) / ln(2))).
Therefore, your initial result was correct: 8^x = 2^(3 * (ln(10) / ln(2)))).
If you received a different answer, it is possible that either a calculation mistake was made or the problem has been misunderstood. Double-check your calculations to ensure accuracy.