4 ^ (3x–1) = 1.5625 × 10 ^(–2)

(3x-1) log4 = log .015625

3x-1 = log.015625/log4
x = (1 + log.015625/log4)/3 = -2/3

Wow, that's a math problem that's burning brighter than a supernova! Let's dive into it, shall we?

First, we need to rewrite that scientific notation to its full form. So, 1.5625 × 10^(-2) becomes 0.015625.

Now, we have the equation 4^(3x-1) = 0.015625. By taking the logarithm of both sides, we can easily solve for x. But hey, who needs logarithms when we can add some humor?

Well, x is like the elusive clown car in a circus - it's hiding somewhere, waiting to be found. To find it, we can break down the equation further. Let's rewrite 0.015625 as (1/64) because, you know, sometimes fractions can be friendlier than decimals.

So, we have the equation 4^(3x-1) = 1/64. Now, let's raise 4 to the power of something to get 1/64. Hmm, it seems like a perfect setup for a clown trick!

Oh, I see! If we rewrite 1/64 as 4^(-3), it's like saying a mouse is as big as an elephant. So, 4^(3x-1) = 4^(-3). Since the bases are the same, we can set the exponents equal to each other.

So, 3x - 1 = -3. Now, let's bring the clown car out of hiding and solve this equation!

If we add 1 to both sides, we get 3x = -2. Divide both sides by 3, and we find that x = -2/3.

And there you have it! The solution is x = -2/3, just like the punchline of a clown's hilarious joke.

To solve the equation 4^(3x-1) = 1.5625 × 10^(-2), we can start by converting the exponential form of the right side to standard decimal form.

1.5625 × 10^(-2) = 0.015625

Now our equation becomes:

4^(3x-1) = 0.015625

Next, we can convert the left side of the equation to the same base.

4^(3x-1) = (2^2)^(3x-1) = 2^(6x-2)

Now our equation becomes:

2^(6x-2) = 0.015625

To get rid of the base 2, we can take the logarithm of both sides. Let's take the logarithm base 2, denoted as log2, of both sides:

log2(2^(6x-2)) = log2(0.015625)

Using the logarithmic property, we can bring down the exponent:

(6x-2)log2(2) = log2(0.015625)

Since log2(2) is equal to 1, we can simplify the equation:

6x - 2 = log2(0.015625)

Now, we solve for x by isolating the variable:

6x = log2(0.015625) + 2

6x = -6 + 2

6x = -4

x = -4/6

x = -2/3

Therefore, the solution to the equation 4^(3x-1) = 1.5625 × 10^(-2) is x = -2/3.

To find the value of x in the equation 4^(3x-1) = 1.5625 × 10^(-2), we can take the logarithm of both sides of the equation. The most common logarithm used is the natural logarithm, denoted as ln.

By taking the natural logarithm of both sides, the equation becomes:

ln(4^(3x-1)) = ln(1.5625 × 10^(-2))

Using the logarithmic property, we can bring down the exponent:

(3x-1) * ln(4) = ln(1.5625 × 10^(-2))

Now, divide both sides of the equation by ln(4):

(3x-1) = ln(1.5625 × 10^(-2)) / ln(4)

Calculate the right side of the equation:

(3x-1) ≈ -3.220 / 1.386

Simplify:

3x - 1 ≈ -2.324

Now, add 1 to both sides of the equation:

3x ≈ -2.324 + 1

3x ≈ -1.324

Finally, divide both sides by 3 to isolate x:

x ≈ -1.324 / 3

x ≈ -0.441

Therefore, the approximate value of x in the equation 4^(3x-1) = 1.5625 × 10^(-2) is x ≈ -0.441.