How to convert a number to $10^n$?

Question

How to convert a number to $10^n$?

For example, $9.3\times10^{-6}=10^{-5.03}$.

How do you do this conversion to calculate the exponent of $10$?

How does one calculate the $-5.03$ exponent?

Answer 1

What you are basically saying is that you have a number $a$ and you want to find some number $x$ such that

$$10^x = a.$$

You can solve the equation above by applying the $\log_{10}$ function on both sides to get

$$\log_{10}(10^x) = \log_{10}(a).$$

Now, remember what the $\log$ function is, and you are done.

Answer 2

If it is in form of $a\times 10^b$, then it is $10^x$ where $x=\log_{10}(a)+b$.

Answer 3

Given the equation:

$10^{n}=9.3\times10^{-6}$

Take $\log_{10}$ on each side:

$n=\log_{10}(9.3\times10^{-6})$

Simplify the right-hand side:

$n=\log_{10}(9.3)+\log_{10}(10^{-6})$

Simplify the right-hand side again:

$n=\log_{10}(9.3)+(-6)$

Solve the right-hand side in a calculator:

$n\approx-5.03$

Answer 4

Well... logarithm values are not easy to hand-calculate, if that's what you're asking. Because $x = 10^{\log_{10} x}$ is the simple definition of logarithm values.

Which is why, once upon a time, kids like me were taught how to look up the relevant values (for $9.3$, in your case) in a table of logarithms: $0.9685$, and then calculate the final logarithm by adjusting for the power of ten. For ease of calculation in later steps, the integer value was sometimes shown negative with the fractional part positive - so your answer there would be $\bar 6.9685$, meaning $-6+0.9685 = -5.0315$ .

Answer 5

Just put it in logarithmic form, since expression is $a^n$.

$\log _{10}(x) = n$

$x = 10^n$

where

$n = \log_{10}(9,3 \times 10^{-6}) = x$