Question

Be sure to read all directions Be neat and precise. Indicate clearly your work as well as on the accuracy simplest form or as red Solve the equation for x without usi log_(8)(x+6)=(2)/(3)

Answer 1

To solve the equation log_8(x+6) = 2/3, transform it into its exponential form as 8^(2/3) = x + 6, simplify to 4 = x + 6, and finally solve for x to get x = -2.

Step-by-step explanation:

To solve the equation log8(x+6) = 2/3, we will use the property of logarithms that equates the log of a number to a power. We know that logb(a) = c is equivalent to bc = a. So, applying this property, we can transform the given logarithmic equation into its exponential form:

To eliminate the log, we write the equation as: 82/3 = x + 6.

Now, we simplify the left side. The exponent 2/3 means we take the cube root of 8, which is 2, and then raise that to the power of 2, which gives us 4: 4 = x + 6.

Next, we isolate x by subtracting 6 from both sides: x = 4 - 6, which simplifies to x = -2.

Finally, we check the answer to see if it is reasonable. Since log8(-2 + 6) is valid and the base 8 with any positive exponent will always yield a positive result, our solution x = -2 is reasonable.