x^(log10x)=100
I'm not sure how to do this at all...
I will assume you meant ...
x^(log10 x = 100
then by definition:
logx 100 = log10x
changing to a common base of the log ...
log10 100/logx = logx/log1010
2/log10x = log10 x/1
(log10 x)^2 = 2
log10 x =√2
using my calculator
x = appr 25.9546 or x = appr .03853
(both answers can be verified)
To solve the equation x^(log10x) = 100, we need to isolate the variable x. Here's a step-by-step explanation of how to approach this problem:
Step 1: Replace log10x with an equivalent expression in terms of base 10 logarithm.
Since logbx = logcx / logcb, we can rewrite log10x as logx / log10.
The equation becomes x^(logx / log10) = 100.
Step 2: Simplify the equation.
Using the property (a^b)^c = a^(b*c), we can rewrite the left side of the equation as (x^logx)^(1 / log10) = 100.
Step 3: Eliminate the exponent to solve for x.
Since (x^logx)^(1 / log10) equals x^(logx / log10), we get x^(logx / log10) = 100. Now we can equate the exponents:
logx / log10 = logx = 2, since x^2 = 100.
Step 4: Solve for x.
To solve for x, we need to convert the equation into exponential form. This means rewriting logx = 2 as x = 10^2.
Therefore, x is equal to 10^2, which gives us x = 100.