Problem:

Find the value of x if (2^(3x-1) - 10) ÷ 7 = 6.

Solution:

To find the value of x, we need to solve the given equation step by step. Let's go through the solution process.

Step 1: Distribute the exponent:

First, let's distribute the exponent of 3x-1 to both terms inside the parentheses. According to the exponent rule, when we have a power raised to another power, we multiply the exponents.

So, (2^(3x-1) - 10) can be written as (2^3x * 2^(-1) - 10).

Step 2: Simplify the exponent:

Since 2^(-1) is the reciprocal of 2^1, we can simplify it as 1/2.

Therefore, the expression becomes (2^3x * 1/2 - 10).

Step 3: Simplify the expression:

Now, let's simplify the expression further by multiplying 2^3x and 1/2.

To multiply two terms with the same base, we add the exponents. Therefore, 2^3x * 1/2 can be written as 2^(3x+1).

The expression now becomes (2^(3x+1) - 10).

Step 4: Solve the equation:

Now that we have simplified the expression, we can solve the equation.

According to the equation, (2^(3x+1) - 10) ÷ 7 = 6.

To isolate the variable, we can multiply both sides of the equation by 7.

7 * (2^(3x+1) - 10) ÷ 7 = 6 * 7.

This simplifies to (2^(3x+1) - 10) = 42.

Step 5: Solve for x:

To solve for x, we can add 10 to both sides of the equation.

(2^(3x+1) - 10) + 10 = 42 + 10.

This simplifies to 2^(3x+1) = 52.

Step 6: Solve the exponent equation:

To solve the exponent equation, we need to take the logarithm of both sides.

Taking the logarithm base 2, we get log2(2^(3x+1)) = log2(52).

The logarithm base 2 cancels out the exponent, and we are left with 3x+1 = log2(52).

Step 7: Solve for x:

To solve for x, we can subtract 1 from both sides of the equation.

3x+1 - 1 = log2(52) - 1.

This simplifies to 3x = log2(52) - 1.

Finally, we can divide both sides of the equation by 3 to isolate x.

3x/3 = (log2(52) - 1)/3.

This gives us the solution x = (log2(52) - 1)/3.

Therefore, x is equal to (log2(52) - 1)/3.