Question Video: Computing Numerical Logarithmic Expressions Using Laws of Logarithms Mathematics • Second Year of Secondary School

Video Transcript

Calculate two log four plus seven log 13, giving you answer to the nearest thousandth.

We could type our expression directly into the calculator. However, in this case, we will simplify the expression first using our laws of logarithms. One of the laws of logarithms states that 𝑛 log 𝑎 is equal to log 𝑎 to the power of 𝑛. The number that we are multiplying the logarithm by becomes the exponent or power. This means that our first term, two log four, can be rewritten as log of four squared. In the same way, seven log 13 is the same as log 13 to the power of seven.

Another one of our laws of logarithms states that log 𝑎 plus log 𝑏 is equal to log of 𝑎𝑏. Our expression can therefore be simplified to log of four squared multiplied by 13 to the power of seven. Whilst we could calculate four squared multiplied by 13 to the power of seven, as 13 to the power of seven is so large, we will leave it in this form in this question. Typing this into the calculator gives us 9.001723 and so on.

We’re asked to round to the nearest thousandth. This is the same as rounding to three decimal places. The seven becomes the deciding number. And if the deciding number is five or greater, we round up. Two log four plus seven log 13 to the nearest thousandth is 9.002.

Note that in this question, there was no base number in either of our terms. If this is the case, we assume that it is log of base 10 and use the log button on our calculator. 10 to the power of 9.002 would be equal to two log four plus seven log 13.