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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?

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4 Answers

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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 $b^x=a$.

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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

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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.

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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.

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