How to get exponent of power if base is known?

Sorry for this type of question, but I've forgotten the math basic from middle school, maybe someone can help me out. If I know the result and base, how can I calculate exponent?

$$2.5 = 10^x$$, how would I get the $$x$$ value of this?

Do you remember the definition of logarithm? That's exactly what you need because the logarithm is defined as the inverse operation of exponentiation (raising a number to a power):

$$2.5=10^x \Longleftrightarrow x=\log_{10}{2.5}.$$

The statement $$x=\log_{b}{a}$$ is asking the question what power should I raise $$b$$ to get $$a$$? And that's equivalent to saying $$b^x=a$$.

The other answers lead you to $$x=\log_{10}(2.5) \approx 0.3979$$ though it is possible to get a close mental arithmetic approximation without explicitly using logarithms:

• $$2^{10}=1024 \approx 1000 = 10^3$$, so $$2 \approx 10^{3/10}$$ or slightly more

• $$2.5 = \dfrac{10}{2^2} \approx \dfrac{10^1}{10^{6/10}}=10^{2/5}$$ or slightly less

• making $$x \approx \dfrac{2}{5} = 0.4$$ or slightly less

You can do something similar for other values, and this can give reasonably good approximations for some other $$\log_{10}(x)$$

x       log_10(x)   approx
1       0           0
1.25    0.09691     0.1-
1.6     0.20412     0.2+
2       0.30103     0.3+
2.5     0.39794     0.4-
3.125   0.49485     0.5-
3.2     0.50515     0.5+
4       0.60206     0.6+
5       0.69897     0.7-
6.25    0.79588     0.8-
8       0.90309     0.9+
10      1           1


Hint: Use the natural logarithm on both sides of the equation.

$$\ln 2.5 = x \ln 10$$ $$\implies x = \dfrac{\ln 2.5}{\ln 10}$$

I used the logarithm law $$\log_a b^r = r\log_a b$$ on the right-hand side of the first equation.

If you know the base and the exponent, then the operation to get the value of the power is exponentiation. For instance, if the base is $$10$$ and the exponent is $$3$$, then the value of the power is $$10^3 = 1000$$.

If you know the power and the exponent, then the operation to get the base is root. For instance, if the exponent is $$3$$ and the power is $$1000$$, then the base is $$\sqrt[3]{1000} = 10$$

If you know the power and the base, the operation to get the exponent is the logarithm. For instance, if the power is $$1000$$ and the base is $$10$$, then the exponent is $$\log_{10}(1000) = 3$$.

These three operations are so closely related, yet their names and notations (and teaching methods) are entirely different. This is a shame, but there is little one can realistically do about it.

In this case, the answer you're looking for is $$\log_{10}(2.5)$$. (Make sure you use logarithms base $$10$$, as $$10$$ is the base of the power. On some calculators, for instance, logarithm base $$10$$ is denoted by $$\log$$ while logarithm base $$e\approx 2.72$$ is denoted by $$\ln$$. But some times $$\log$$ refers to base $$e$$ logarithms instead. You will have to test this on your calculator to figure out which convention it follows.