What is the exact value of x?
5⋅84x=376
x=log75.2/4log8
x=4log8/log75.2 <my answer
x=log84/log75.2
x=4log75.2/log8
not quite...
5⋅8^(4x) = 376
8^(4x) = 75.2
4x log8 = log75.2
x = log75.2/(4log8)
or, since 4log8 = log4096,
x = log409675.2
To find the exact value of x in the equation 5⋅84x=376, you can use logarithms.
Step 1: Write the equation in logarithmic form:
84x = log376 / log5
Step 2: Use the change of base formula to convert the base from 84 to a common logarithmic base (such as 10):
x = log(log376 / log5) / log84
So, the exact value of x is x = log(log376 / log5) / log84.
To find the value of x in the equation 5⋅84x = 376, we can use logarithms.
First, let's rewrite the equation in logarithmic form. Taking the logarithm (base 8) of both sides of the equation gives:
log8(5⋅84x) = log8(376)
Now let's apply the rules of logarithms. Using the power rule, we can move the exponent down:
log8(5) + log8(84x) = log8(376)
Next, we simplify the logarithmic expressions. Since log8(5) is a constant, we can treat it as a single term:
C=log8(5)
Now the equation becomes:
C + log8(84x) = log8(376)
To isolate the 84x term, we subtract C from both sides:
log8(84x) = log8(376) - C
Now, we can use a property of logarithms to get rid of the logarithm on the left side. When we have logb(x) = logb(y), the bases are the same, so the arguments (x and y) must be equal:
84x = 376 - C
To find the value of C, we need to calculate log8(5). We can use a calculator to find the logarithm base 8 of 5:
C = log8(5) ≈ 0.6931
Now we substitute the value of C back into the equation:
84x = 376 - 0.6931
Simplifying further:
84x ≈ 375.307
Finally, to solve for x, divide both sides of the equation by 84:
x ≈ 4.466
Therefore, the approximate value of x in the equation 5⋅84x = 376 is 4.466.