Video Transcript
Use a calculator to find the value
of 𝑥 for which two to the power of 𝑥 multiplied by seven is equal to 16 multiplied
by seven to the power of 𝑥 plus nine. Give your answer correct to two
decimal places.
We will begin this question by
taking logs of both sides of the equation. This gives us log of two to the
power of 𝑥 multiplied by seven is equal to log of 16 multiplied by seven to the
power of 𝑥 plus nine. We recall that one of our laws of
logarithms states that log 𝑥 plus log 𝑦 is equal to log of 𝑥 multiplied by
𝑦. This means we can rewrite the
left-hand side as log two to the power of 𝑥 plus log of seven. The left-hand side can be rewritten
as log 16 plus log seven to the power of 𝑥 plus nine.
One of our other laws of logarithms
states that log 𝑥 to the power of 𝑛 is equal to 𝑛 log 𝑥. We can rewrite the first term on
the left-hand side as 𝑥 log two. The final term on the right-hand
side can be rewritten as 𝑥 plus nine multiplied by log seven. We can distribute our parentheses
here to get 𝑥 log seven plus nine log seven.
Our equation has therefore become
𝑥 log two plus log seven is equal to log 16 plus 𝑥 log seven plus nine log
seven. Two of our five terms have an 𝑥 in
them, and we need to ensure that these are on the same side of the equation. Subtracting log seven and 𝑥 log
seven from both sides of the equation gives us 𝑥 log two minus 𝑥 log seven is
equal to log 16 plus nine log seven minus log seven. The last two terms on the
right-hand side can be simplified. Nine log seven minus log seven is
equal to eight log seven.
We can then factor out the 𝑥 on
the left-hand side so that we get 𝑥 multiplied by log two minus log seven. This is equal to log 16 plus eight
log seven. Finally, we divide both sides by
log two minus log seven. Typing this into the calculator, we
get 𝑥 is equal to negative 14.6395 and so on. We are asked to round our answer to
two decimal places. Therefore, 𝑥 is equal to negative
14.64. This is the value for 𝑥 for which
two to the power of 𝑥 multiplied by seven is equal to 16 multiplied by seven to the
power of 𝑥 plus nine.
