solve each exponential equation. Given the value for x
8^x=1/64
3^x=6
Can someone show me how to preform these. I would greatly appreciate it
Thank You
log ( a ^ n ) = n * log ( a )
log ( 1 / a ) = - log ( a )
log ( a * b ) = log ( a ) + log ( b )
64 = 8 * 8
log ( 64 ) = log ( 8 ) + log ( 8 )
log ( 64 ) = 2 * log ( 8 )
8 ^ x = 1 / 64
x * log ( 8 ) = log ( 1 / 64 )
x * log ( 8 ) = - log ( 8 * 8 )
x * log ( 8 ) = - [ log ( 8 ) +
log ( 8 ) ]
x * log ( 8 ) = - [ 2 * log ( 8 ) ]
x * log ( 8 ) = - 2 * log ( 8 ) Divide both sides by log ( 8 )
X = - 2
6 = 3 * 2
log ( 6 ) = log ( 3 ) + log ( 2 )
3 ^ x = 6
x * log ( 3 ) = log ( 6 )
x * log ( 3 ) = log ( 3 ) + log ( 2 ) Divide both sides by log ( 3 )
x = log ( 3 ) / log ( 3 ) + log ( 2 ) / log ( 3 )
x = 1 + log ( 2 ) / log ( 3 ) = 1.6309
Hi Bosnian,
I want thank you for working these problems out for me. I see how to work out the problems. You are great. I was getting so stressed, but now you have given me the faith that I can do this. I went back to school at an older age and algebra was taken when I was in 8th grade. Thank you for your time and I appreciate it so much
You're welcome.
Certainly! I can help you solve these exponential equations step by step.
1. 8^x = 1/64
To solve this equation, we need to find the exponent (x) that satisfies the equation. Notice that the base 8 can be written as 2^3. So we can rewrite the equation as:
(2^3)^x = 1/64
Applying the power of a power rule, we get:
2^(3x) = 1/64
To simplify further, we can rewrite 1/64 as 2^-6 (since 64 = 2^6):
2^(3x) = 2^-6
Since the bases are the same, we can equate the exponents:
3x = -6
Now divide both sides by 3:
x = -6/3
Simplifying further, we have:
x = -2
Therefore, the solution to the equation 8^x = 1/64 is x = -2.
2. 3^x = 6
To solve this equation, we need to find the exponent (x) that satisfies the equation. Unfortunately, there is no simple way to rewrite 6 as a power of 3. Therefore, we'll need to use logarithms to solve for x.
Take the logarithm of both sides of the equation using the common logarithm (log base 10) or the natural logarithm (ln):
log(3^x) = log(6)
This can also be written as:
x * log(3) = log(6)
Now divide both sides by log(3):
x = log(6) / log(3)
Using a calculator, you can evaluate the expression on the right-hand side to get the approximate value of x.
Therefore, the solution to the equation 3^x = 6 is x ≈ 1.6309 (rounded to four decimal places).
I hope this helps! Let me know if you have any further questions.