Question Video: Finding the Solution Set of Exponential Equations | Nagwa Question Video: Finding the Solution Set of Exponential Equations | Nagwa

# Question Video: Finding the Solution Set of Exponential Equations Mathematics • Second Year of Secondary School

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Use a calculator to find the value of 𝑥 for which 2^𝑥 × 7 = 16 × 7^(𝑥 + 9). Give your answer correct to two decimal places.

03:09

### 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.

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