I need help with the following problems:
~23. 3^(x-6)=81 How do you solve this? I'm not sure what becomes of the -6.
~24. 11-6^x=3
(x-6) log 3 = log 8
x-6 = log 8 / log 3
x = 6 + (log 8/log 3)
Sure! Let's solve these problems step by step.
Problem #23: 3^(x-6) = 81
To solve this equation, we need to get rid of the exponent on the left side. Since 81 is equal to 3^4, we can rewrite the equation as:
3^(x-6) = 3^4
Now, we can set the exponents equal to each other:
x - 6 = 4
To isolate x, we can add 6 to both sides of the equation:
x = 4 + 6
Simplifying that:
x = 10
So the solution to the equation is x = 10.
Problem #24: 11 - 6^x = 3
In this equation, we also need to get rid of the exponent, but this time the exponent is being applied to a base number of 6. To do that, we need to take the logarithm of both sides of the equation.
We can take logarithm base 6 of both sides:
log6(11 - 6^x) = log6(3)
Now, using the logarithmic property, we can bring down the exponent:
xlog6(6) - log6(11) = log6(3)
Since log6(6) is equal to 1, we can simplify further:
x - log6(11) = log6(3)
To isolate x, we can add log6(11) to both sides of the equation:
x = log6(3) + log6(11)
So the solution to the equation is x = log6(3) + log6(11).
Remember to check the solutions to see if they satisfy the original equations.