# Rewrite as a sum or difference of multiple logarithms solver

I'll do this in blue. It is possible to give similar proofs that the other index laws also hold for negative integer and rational exponents.

### Rewriting logarithms

To round a number to a required number of significant figures, first write the number in scientific notation and identify the last significant digit required. So we have to raise 5 to the second power to get to Since all measurements are approximations anyway, they generally report the numbers rounded to a given number of significant figures. Hence, for example, the diameter of a uranium atom is 0. Thus, 0. Log base 5 of 25 to the x over y using this property means that it's the same thing as log base 5 of 25 to the x power minus log base 5 of y. The star Sirius is approximately 75 km from the sun. It seems like the relevant logarithm property here is if I have log base x of a to the b power, that's the same thing as b times log base x of a, that this exponent over here can be moved out front, which is what we did it right over there. And here we have 25 to the x over y. Significant figures in scientific notation Scientists and engineers routinely employ scientific notation to represent large and small numbers. If we move the decimal point 13 places to the right, inserting the necessary zeroes, we arrive back at the number we started with. So we can simplify. We place the decimal point just after the first non-zero digit and multiply by the appropriate power of ten. This is the same as rounding the number 21 to 21 , that is, correct to three significant figures.

So we can use some logarithm properties. To round a number to a required number of significant figures, first write the number in scientific notation and identify the last significant digit required.

Solution 3. So let me write this down. We can similarly deal with very small numbers using negative indices.

### Log(a+b)

Solution A table of approximate values follows: x. And I do agree that this does require some simplification over here, that having this right over here inside of the logarithm is not a pleasant thing to look at. Since all measurements are approximations anyway, they generally report the numbers rounded to a given number of significant figures. And then, of course, we have minus log base 5 of y. Significant figures in scientific notation Scientists and engineers routinely employ scientific notation to represent large and small numbers. Properties of logarithms Video transcript We're asked to simplify log base 5 of 25 to the x power over y. If we move the decimal point 13 places to the right, inserting the necessary zeroes, we arrive back at the number we started with. Solution 3. This is the same as rounding the number 21 to 21 , that is, correct to three significant figures. To round a number to a required number of significant figures, first write the number in scientific notation and identify the last significant digit required. So let me write this down. Now, this looks like we can do a little bit of simplifying. And we're done. A given number may be expressed with different numbers of significant figures. We can represent this number more compactly by moving the decimal point to just after the first non-zero digit and multiplying by an appropriate power of 10 to recover the original number.

Properties of logarithms Video transcript We're asked to simplify log base 5 of 25 to the x power over y.

For example, 3. Thus, 0. So then we are left with, this is equal to-- and I'll write it in front of the x now-- 2 times x minus log base 5 of y. We place the decimal point just after the first non-zero digit and multiply by the appropriate power of ten.

## Log base properties

Log base 5 of 25 to the x over y using this property means that it's the same thing as log base 5 of 25 to the x power minus log base 5 of y. In this case, we could leave this as the answer, or, if required, write is as 4. So we can simplify. Significant figures in scientific notation Scientists and engineers routinely employ scientific notation to represent large and small numbers. For example, 3. And then, of course, we have minus log base 5 of y. And I do agree that this does require some simplification over here, that having this right over here inside of the logarithm is not a pleasant thing to look at. I'll do this in blue.

It seems like the relevant logarithm property here is if I have log base x of a to the b power, that's the same thing as b times log base x of a, that this exponent over here can be moved out front, which is what we did it right over there. So this simplifies to 2.

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