Question

Solve the logarithmic equation algebre log(3z)=2 z

Answer 1

The solution of the logarithmic equation algebre log(3z)=2 z is z = (10^2)/3.

Step-by-step explanation:

To solve the logarithmic equation algebre log(3z) = 2z, we need to follow these steps:

1. Isolate the variable z:

First, we need to isolate the variable z on one side of the equation. To do this, we can use the following property of logarithms: log(a^m) = m * log(a). Applying this property, we get:

2z = log(3z)

Dividing both sides by 2, we get:

z = (1/2) * log(3z)

Now, we can use the following property of logarithms: log(x^m) = m * log(x). Applying this property, we get:

z = (1/2) * log(3) + (1/2) * log(z)

Next, we can separate the terms containing z and simplify them:

z = (1/2) * [log(3) + log(z)]

Now, we can use the following property of logarithms: log(xy) = log(x) + log(y). Applying this property, we get:

z = (1/2) * [log(3z)] = (1/2) * [log(3) + log(z)] = (1/2) * [log(3) + z * log(3)] = (1/2) * [log(3)] + (1/2) * [z * log(3)]

Now, let's define a new variable x = z * log(3). Substituting this in the above equation, we get:

z = (1/2) * [log(3)] + (x / 2) / [log(3)]

Now, let's solve for x in terms of z:

x = z * log(3), and substituting this in the above equation, we get:

z = (1/2) * [log(3)] + (z * log(3)) / [2 * log(3)]

Now, let's simplify the expression inside the square brackets:

[log(3)] / [2 * log(3)] = 1 / 2. Substituting this in the above equation, we get:

z = (1/2) * z + (1/2), which simplifies to:

z - (1/2) = (1/2) * z. Now, let's isolate z on one side of the equation:

z - (1/2) = (1/2) * z. Adding and subtracting (1/2), we get:

z - (1/2) + (1/2) = (1/2) * z + (1/2). Now, let's simplify both sides of the equation:

z = z. Since both sides are equal to each other, our solution is unique and is given by:

z = (10^2)/3. This is our final answer.

Answer 2

The solution is
`\[ z = (1)/(9) \]`.

Step-by-step explanation:

The given logarithmic equation is
`\( \log(3z) = 2z \). To solve for \( z \)`, we need to eliminate the logarithm. Begin by expressing the logarithmic equation in exponential form. The base of the logarithm is 10 by default, so:

`\( 10^(2z) = 3z \)`.

Now, rearrange the equation to isolate
`\( z \): \( 100 = 3z \).`

Finally, divide both sides by 3 to find the value of
`\( z \): \( z = (1)/(9) \).`

In the logarithmic equation
`\( \log(3z) = 2z \)`,

we used the property
`\( \log_a(b) = c \) is equivalent to \( a^c = b \)`.

By applying this property, we transformed the equation into
`\( 10^(2z) = 3z \).`

This step is crucial to simplify the equation and make it more manageable. The subsequent algebraic manipulations involved rearranging terms to isolate
`\( z \)` on one side of the equation.

The solution
`\( z = (1)/(9) \)`was obtained after dividing both sides by 3.

Therefore,
`\( z = (1)/(9) \)` satisfies the original logarithmic equation.

In summary, solving the logarithmic equation involved converting it into exponential form and performing algebraic operations to find the value of
`\( z \)`. The final solution
`\( z = (1)/(9) \)` is consistent with the mathematical principles applied throughout the solution process.