Solve for y natural log of 2y-5+ natural log of 2=5x+ natural log of 5x
The given problem is a logarithmic equation that involves solving for the variable y. The equation includes natural logarithms (logarithms with base e) of expressions involving y and constants. Specifically, the terms within the natural logarithms are (2y - 5) and 2 on the left side of the equation, and 5x and 5x on the right side of the equation. Solving this question would involve using properties of logarithms, such as the product, quotient, or power properties, to combine or separate the logarithmic terms, and then possibly utilizing exponentiation to remove the logarithms entirely and solve for y in terms of the other variables and constants present in the equation.
$ln \left(\right. 2 y - 5 \left.\right) + ln \left(\right. 2 \left.\right) = 5 x + ln \left(\right. 5 x \left.\right)$
Answer
Solution:
Step 1: Simplify the equation's left-hand side.
Step 1.1: Utilize the logarithmic product rule: $\ln(a) + \ln(b) = \ln(a \cdot b)$. Combine the logs on the left side: $\ln((2y - 5) \cdot 2) = 5x + \ln(5x)$.
Step 1.2: Distribute the multiplication across the sum inside the log: $\ln(2 \cdot 2y - 2 \cdot 5) = 5x + \ln(5x)$.
Step 1.3: Perform the multiplication:
Step 1.3.1: Multiply $2$ by $2y$: $\ln(4y - 2 \cdot 5) = 5x + \ln(5x)$.
Step 1.3.2: Multiply $2$ by $5$: $\ln(4y - 10) = 5x + \ln(5x)$.
Step 2: Isolate the logarithmic terms.
- Move the log term from the right to the left side: $\ln(4y - 10) - \ln(5x) = 5x$.
Step 3: Apply the logarithmic quotient rule.
- Use the rule $\ln(a) - \ln(b) = \ln(\frac{a}{b})$: $\ln\left(\frac{4y - 10}{5x}\right) = 5x$.
Step 4: Factor out common terms.
Step 4.1: Factor out a $2$ from $4y - 10$:
Step 4.2: Factor $2$ from $4y$: $\ln\left(\frac{2(2y) - 10}{5x}\right) = 5x$.
Step 4.3: Factor $2$ from $-10$: $\ln\left(\frac{2(2y) + 2(-5)}{5x}\right) = 5x$.
Step 4.4: Factor $2$ from the entire expression: $\ln\left(\frac{2(2y - 5)}{5x}\right) = 5x$.
Step 5: Convert the log equation to an exponential equation.
- Apply the exponential function to both sides: $e^{\ln\left(\frac{2(2y - 5)}{5x}\right)} = e^{5x}$.
Step 6: Use the definition of logarithms to simplify.
- The equation $e^{\ln(x)} = x$ simplifies to: $\frac{2(2y - 5)}{5x} = e^{5x}$.
Step 7: Solve for $y$.
Step 7.1: Multiply both sides by $5x$: $2(2y - 5) = 5x \cdot e^{5x}$.
Step 7.2: Distribute and simplify:
Step 7.2.1: Multiply $2$ by $2y$ and $-5$: $4y - 10 = 5x \cdot e^{5x}$.
Step 7.2.2: Add $10$ to both sides: $4y = 5x \cdot e^{5x} + 10$.
Step 7.2.3: Divide by $4$: $y = \frac{5x \cdot e^{5x}}{4} + \frac{10}{4}$.
Step 7.2.4: Simplify the fraction: $y = \frac{5x \cdot e^{5x}}{4} + \frac{5}{2}$.
Knowledge Notes:
Logarithmic Properties:
Product Rule: $\ln(ab) = \ln(a) + \ln(b)$.
Quotient Rule: $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$.
Power Rule: $\ln(a^b) = b\ln(a)$ (not used in this solution).
Change of Base Formula: $\log_b(a) = \frac{\ln(a)}{\ln(b)}$ (not used in this solution).
Logarithm Definition: If $b^y = x$ and $b > 0$, then $y = \log_b(x)$.
Exponential and Logarithmic Relationship: The natural logarithm $\ln(x)$ is the inverse function of the exponential function $e^x$. Therefore, $e^{\ln(x)} = x$ and $\ln(e^x) = x$.
Distributive Property: $a(b + c) = ab + ac$. This property is used to expand or factor algebraic expressions.
Isolating Variables: To solve for a variable, we manipulate the equation to get the variable on one side and all other terms on the other side, using inverse operations like addition/subtraction and multiplication/division.
Combining Like Terms: When solving equations, we combine terms that have the same variable raised to the same power.
Factoring: This involves finding a common factor in terms and pulling it out of the expression, simplifying the equation and making it easier to solve.
