Solve for y natural log of 2y-5+ natural log of 2=5x+ natural log of 5x

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:

1. 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 $y={\log }_b(x)$.

2. 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 $\ln (e^x)=x$.

3. Distributive Property: $a(b+c)=ab+ac$. This property is used to expand or factor algebraic expressions.

4. 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.

5. Combining Like Terms: When solving equations, we combine terms that have the same variable raised to the same power.

6. 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.