A short problem

Algebra Level 5

Find the value of $a$ for which following equation has only one solution:

$\large a^x=\log_a{x}$

The answer is 1.44471.

1 solution

Brandon Monsen
Dec 7, 2015

I don't know an algebraic way to show the answer, but I used calculus. The functions $a^{x}$ and $log_{a}(x)$ are symmetric about the line $y=x$ . This means we want the maximum value of $a$ such that $a^{x}=x$ . (I actually posted a problem very similar to this).

$a^{x}=x \\ a=x^{\frac{1}{x}} \\a=e^{\frac{1}{x}\ln(x)} \\ \frac{da}{dx}=e^{\frac{1}{x}ln(x)}(\frac{1}{x} \times \frac{1}{x}+\frac{-1}{x^{2}} \times ln(x))$

To find maximums, we want $\frac{da}{dx}=0$ , so in other words we want $\ln(x)=1$ , so $x=e$ .

This means $a=e^{\frac{1}{e}} \approx \boxed{1.44}$

Isn't a=1 a solution too? The question should be reworded asking the maximum a ar which this happens

Pratyush Pandey - 4 years, 3 months ago

@Brandon Monsen Why the maximum value?

Axas Bit - 5 years, 5 months ago

Since $log_{a}(x)$ and $a^{x}$ are symmetric about the line $y=x$ , we know that if the two functions have only one solution, then one of the two is tangent to the line $y=x$ . This means we are looking for the maximum value of $a$ such that $a^{x}=x$ has at least one solution. If we make $a$ any greater, the two lines will never cross. It's difficult to explain in words but playing around with $a^{x}$ on desmos would be a great place to see for yourself.

Brandon Monsen - 5 years, 5 months ago

A short problem

The answer is 1.44471.

1 solution

0 pending reports