If log base 9 of 2 = 3x+1, find x to the nearest tenth.
log 9 2 = 3x+1
then
9^(3x+1) = 2
log both sides
(3x+1)log9 = log2
3x+1 = log2/log9
3x = log2/log9 - 1
x = (log2/log9 - 1)/3
you do the button-pushing.
To find the value of x in the equation log base 9 of 2 = 3x + 1, we have to apply the properties of logarithms. Here's how we can solve it step by step:
Step 1: Start with the equation log base 9 of 2 = 3x + 1.
Step 2: Rewrite the equation using the definition of logarithms: 9^(3x + 1) = 2.
Step 3: Rewrite 9 as 3^2: (3^2)^(3x + 1) = 2.
Step 4: Apply the exponent rule (a^m)^n = a^(m*n): 3^(2(3x + 1)) = 2.
Step 5: Simplify the exponent on the left side of the equation: 3^(6x + 2) = 2.
Step 6: Convert both sides of the equation to exponential form: 2 = 3^(6x + 2).
Step 7: To isolate the variable, take the logarithm of both sides using the base 3: log base 3 of 2 = log base 3 of (3^(6x + 2)).
Step 8: Apply the logarithm property log base b of (b^c) = c: log base 3 of 2 = (6x + 2).
Step 9: Move the constant term to the other side of the equation: log base 3 of 2 - 2 = 6x.
Step 10: Simplify: log base 3 of 2 - 2 = 6x.
Step 11: Divide both sides by 6: (log base 3 of 2 - 2)/6 = x.
Step 12: Use a calculator to approximate the value of (log base 3 of 2 - 2)/6 to the nearest tenth.
By following these steps, you can solve the equation and find the value of x to the nearest tenth.