Solve for k cube root of 5x^(k+1)* cube root of 25x^k=5x^7

Solution:

Step 1:

Apply the natural logarithm to both sides to simplify the exponents.

$$\ln(\sqrt[3]{5x^{k+1}} \cdot \sqrt[3]{25x^k}) = \ln(5x^7)$$

Step 2:

Distribute the logarithm over the product on the left side.

Step 2.1:

Express the left side as a sum of logarithms.

$$\ln(\sqrt[3]{5x^{k+1}}) + \ln(\sqrt[3]{25x^k}) = \ln(5x^7)$$

Step 2.2:

Convert cube roots to exponent form.

$$\ln((5x^{k+1})^{\frac{1}{3}}) + \ln((25x^k)^{\frac{1}{3}}) = \ln(5x^7)$$

Step 2.3:

Apply the power rule of logarithms to move the exponents out front.

$$\frac{1}{3}\ln(5x^{k+1}) + \frac{1}{3}\ln(25x^k) = \ln(5x^7)$$

Step 2.4:

Separate the logarithms of products into sums.

$$\frac{1}{3}(\ln(5) + \ln(x^{k+1})) + \frac{1}{3}(\ln(25) + \ln(x^k)) = \ln(5x^7)$$

Step 2.5:

Apply the power rule of logarithms to the terms with exponents.

$$\frac{1}{3}(\ln(5) + (k+1)\ln(x)) + \frac{1}{3}(\ln(25) + k\ln(x)) = \ln(5x^7)$$

Step 3:

Combine like terms on the left side.

Step 3.1:

Combine the coefficients and logarithms.

$$\frac{1}{3}\ln(5) + \frac{k+1}{3}\ln(x) + \frac{1}{3}\ln(25) + \frac{k}{3}\ln(x) = \ln(5x^7)$$

Step 3.2:

Add the terms involving $k$.

$$\frac{2k+1}{3}\ln(x) + \frac{\ln(5) + \ln(25)}{3} = \ln(5x^7)$$

Step 4:

Multiply through by 3 to clear the denominators.

$$2k\ln(x) + \ln(x) + \ln(5) + \ln(25) = 3\ln(5x^7)$$

Step 5:

Isolate terms with $k$ on one side and constants on the other.

$$2k\ln(x) = 3\ln(5x^7) - \ln(x) - \ln(5) - \ln(25)$$

Step 6:

Solve for $k$ by dividing by $2\ln(x)$.

$$k = \frac{3\ln(5x^7) - \ln(x) - \ln(5) - \ln(25)}{2\ln(x)}$$

Knowledge Notes:

1. Natural Logarithm: The natural logarithm, denoted as $\ln(x)$, is the logarithm to the base $e$, where $e$ is an irrational and transcendental constant approximately equal to 2.71828.

2. Logarithm Properties:

   • Product Rule: $\ln(xy) = \ln(x) + \ln(y)$
   • Quotient Rule: $\ln(\frac{x}{y}) = \ln(x) - \ln(y)$
   • Power Rule: $\ln(x^a) = a\ln(x)$

3. Exponents and Roots: The $n$-th root of $a$ can be written as $a^{\frac{1}{n}}$, so the cube root of $a$ is $a^{\frac{1}{3}}$.

4. Combining Like Terms: When simplifying expressions, terms with the same variable and exponent can be combined.

5. Isolating the Variable: To solve for a variable, we manipulate the equation to get the variable on one side and all other terms on the other side. This often involves using inverse operations like addition/subtraction or multiplication/division.