Is the answer to log sub a 2 = .4, log sub a 3 = .5, log sub a 5 = .8 is it 2.9? Is the answer to log sub 5 (2x+3) = 2 is the correct answer 11? Is the solution to 8^x=16^x+2 is the solution 2?
Are you guessing? Something is wrong. Let me see your work. You are missing something important, I think.
To solve logarithmic equations, we can use the properties of logarithms and algebraic techniques. Let's go through each equation step by step:
1. log_a(2) = 0.4, log_a(3) = 0.5, log_a(5) = 0.8 and we need to find log_a(2.9):
To find log_a(2.9), we can use the property of logarithms which states that log_a(m) + log_a(n) = log_a(m * n). In this case, we have log_a(2) + log_a(3) = log_a(2 * 3) = log_a(6).
Now, we can use the same property to rewrite log_a(6) + log_a(5) = log_a(6 * 5) = log_a(30).
Therefore, log_a(2.9) is equal to log_a(30), not 2.9.
2. log_5(2x + 3) = 2 and we need to find x:
To solve for x, we can use the property of logarithms which states that log_b(m) = n is equivalent to b^n = m.
In this case, we have log_5(2x + 3) = 2. Rewriting this equation, we get 5^2 = 2x + 3, which simplifies to 25 = 2x + 3.
Now, we isolate x by subtracting 3 from both sides: 25 - 3 = 2x, which gives us 22 = 2x.
Finally, we divide both sides by 2 to solve for x: 22/2 = 2x/2, resulting in x = 11.
Therefore, the correct answer for log_5(2x + 3) = 2 is x = 11.
3. 8^x = 16^(x + 2) and we need to find x:
To solve for x, we can use the property of exponents which states that a^(b * c) = (a^b)^c.
In this case, we can rewrite 16 as 8^2 since 16 is equal to 8 squared. Now, the equation becomes 8^x = (8^2)^(x + 2).
Using the property mentioned above, we can simplify this to 8^x = 8^(2 * (x + 2)).
To eliminate the base 8 from both sides, we can equate the exponents: x = 2 * (x + 2).
Expanding the equation, we get x = 2x + 4.
Next, we subtract 2x from both sides to solve for x: x - 2x = 2x + 4 - 2x, which simplifies to -x = 4.
Finally, we multiply both sides by -1 to isolate x: (-1) * (-x) = 4 * (-1), resulting in x = -4.
Therefore, the solution to 8^x = 16^(x + 2) is x = -4.
I apologize if there was any confusion or mistakes in my previous responses. It's important to carefully follow the steps and properties when solving logarithmic or exponential equations.