Use the properties of logarithms to evaluate/simplify the expression.
1. 5^log(7x)
2. 10^x + 2
log(x)=x+2 or log(x)=x-2 ????
To evaluate/simplify the expression 5^log(7x), we can utilize the property of logarithms which states that log_b(b^x) = x.
Here's how you can solve it step by step:
1. Start with the original expression: 5^log(7x).
2. Apply the property of logarithms: 5^log(7x) is equivalent to (7x).
So, the expression 5^log(7x) simplifies to (7x).
Now, let's move on to the second question about the equation log(x) = x + 2 or log(x) = x - 2.
To determine which equation is correct, we need to solve for x in both equations and see which one satisfies the given equation.
For log(x) = x + 2:
1. Start with the original equation: log(x) = x + 2.
2. Convert it to exponential form: 10^(x + 2) = x.
3. Simplify the exponential expression: 10^2 * 10^x = x.
4. Rewrite the equation: 100x = x.
5. Combine like terms and solve for x: 99x = 0, x = 0.
So, x = 0 is a solution to log(x) = x + 2.
Now, let's move on to log(x) = x - 2:
1. Start with the original equation: log(x) = x - 2.
2. Convert it to exponential form: 10^(x - 2) = x.
3. Simplify the exponential expression: 10^(-2) * 10^x = x.
4. Rewrite the equation: (1/100) * 10^x = x.
5. Combine like terms and solve for x: x = 0 is not a solution since it is not possible for x to equal (1/100) * 10^x.
So, log(x) = x - 2 does not have any solutions.
In conclusion, the equation log(x) = x + 2 has a solution of x = 0, whereas log(x) = x - 2 does not have any solutions.