Use a calculator to find the value of 𝑥 for which three to the power of negative four 𝑥 minus three is equal to eight to the power of 𝑥 plus 4.7. Give your answer correct to two decimal places.
In order to calculate part or all of the exponents in an exponential equation like this, we use logarithms. In this question, we begin by taking the log to the base 10 of both sides of the equation. One of our laws of logarithms states that log 𝑎 to the power of 𝑥 is equal to 𝑥 log 𝑎. We can use this to bring down our exponents or powers. The left-hand side of the equation can be rewritten as negative four 𝑥 minus three multiplied by log three. The right-hand side is equal to 𝑥 plus 4.7 multiplied by log eight.
We can then distribute the parentheses, otherwise known as expanding the brackets. This gives us negative four 𝑥 log three minus three log three is equal to 𝑥 log eight plus 4.7 log eight. Two of our four terms contain an 𝑥. We need to get these on one side of the equation. Adding four 𝑥 log three and subtracting 4.7 log eight from both sides gives us negative three log three minus 4.7 log eight is equal to 𝑥 log eight plus four 𝑥 log three. We can then factor or factorize 𝑥 out on the right-hand side. This gives us 𝑥 multiplied by log eight plus four log three.
Dividing both sides by this bracket or parentheses gives us 𝑥 is equal to negative three log three minus 4.7 log eight divided by log eight plus four log three. Typing this into the calculator, we get negative 2.018756 and so on. As we need to give our answer to two decimal places, the eight is the deciding number. When the deciding number is five or greater, we round up. Therefore, 𝑥 is equal to negative 2.02. We can check this answer by substituting our value of 𝑥 into the left- and right-hand side of the initial equation.