find the value of x when:
x= 5 log base 2 (1/16)+ 8 log base 3 (27)
To find the value of x in the given equation:
x = 5 log₂(1/16) + 8 log₃(27)
We can use the properties of logarithms to simplify the expression. Let's work through it step by step:
1. Simplify the logarithmic expressions:
Since 1/16 is equal to 2⁻⁴, we can rewrite the first term as follows:
x = 5 log₂(2⁻⁴) + 8 log₃(27)
The exponent can be brought down in front of the logarithm:
x = -20 log₂(2) + 8 log₃(27)
Since log₂(2) is equal to 1, we have:
x = -20 + 8 log₃(27)
2. Evaluate the logarithmic expressions:
Next, let's simplify log₃(27). Since 27 can be written as 3³:
x = -20 + 8 log₃(3³)
The exponent can be brought down in front of the logarithm:
x = -20 + 8 × 3
x = -20 + 24
x = 4
Therefore, the value of x is 4.