For which value of x is the equation true?
10^x = 2e
A. e/5
B. 2/ln10
C. 1+ln2/ln10
D. 1 + log2
taking natural log ... x ln(10) = ln(2) + 1
I would just take logs of both sides
log 10^x = log (2e)
x log10 = log(2e)
x (1) = log(2e)
x = log(2e) <----- nice and compact, should have been a choice
To find the value of x for which the equation 10^x = 2e is true, we need to solve for x.
First, let's take the natural logarithm (ln) on both sides of the equation to eliminate the exponential function:
ln(10^x) = ln(2e)
Using the property of logarithms that ln(a^b) = b * ln(a), we can simplify the left side of the equation:
x * ln(10) = ln(2e)
Next, we know that ln(10) is approximately 2.3026 and ln(2e) can be simplified as follows:
ln(2e) = ln(2) + ln(e) = ln(2) + 1 (because ln(e) = 1)
Now, we have:
x * 2.3026 = ln(2) + 1
To isolate x, we can subtract 1 from both sides:
x * 2.3026 - 1 = ln(2)
Finally, to solve for x, we divide both sides by 2.3026:
x = (ln(2) - 1) / 2.3026
To determine which option among A, B, C, and D represents this value, we can substitute the given values into the equation:
A. e/5
B. 2/ln10
C. 1 + ln2/ln10
D. 1 + log2
By substituting, we find that option C (1 + ln2/ln10) represents the value of x.
Therefore, the correct answer is C.