13^(2x)-5*3^(x)+6=0

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13^(2x)-5*3^(x)+6=0

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Identify Equation Type: Step Title: Identify the Type of Equation Concise Step Description: Determine the type of equation we are dealing with to apply the appropriate method for solving it. Step Calculation: The equation is 13^(2x) - 5*3^(x) + 6 = 0. This is not a standard quadratic equation due to the different bases of the exponents. However, we can try to transform it into a quadratic form by substitution if possible.Find Substitution: Step Title: Look for a Substitution Concise Step Description: Find a substitution that can transform the equation into a quadratic form. Step Calculation: Let's denote 3^(x) as 'y'. Then, 13^(2x) can be written as (3^(x))^2, which is y^2. The equation now looks like y^2 - 5y + 6 = 0, which is a quadratic equation in terms of 'y'.Factor Quadratic Equation: Step Title: Factor the Quadratic Equation Concise Step Description: Factor the quadratic equation obtained after the substitution. Step Calculation: We need to find two numbers that multiply to 6 (the constant term) and add up to -5 (the coefficient of the 'y' term). The numbers that satisfy these conditions are -2 and -3. Therefore, the factored form is (y - 2)(y - 3) = 0.Solve for Substituted Variable: Step Title: Solve for the Substituted Variable Concise Step Description: Solve the factored equation for the substituted variable 'y'. Step Calculation: We have two possible solutions for 'y': y - 2 = 0 or y - 3 = 0. This gives us y = 2 or y = 3.Back-Substitute for 'x': Step Title: Back-Substitute to Solve for 'x' Concise Step Description: Replace 'y' with 3^(x) and solve for 'x'. Step Calculation: Since y = 3^(x), we have two equations: 3^(x) = 2 and 3^(x) = 3. Solving for 'x' gives us x = log_3(2) and x = log_3(3). The second equation simplifies to x = 1 because 3^(1) = 3.Solve Logarithmic Equations: Step Title: Solve the Logarithmic Equations Concise Step Description: Solve the logarithmic equations to find the values of 'x'. Step Calculation: The first equation is x = log_3(2). This cannot be simplified further without a calculator. The second equation has already been simplified to x = 1.

$13^{2 x}-5 \cdot 3^{x}+6=0$

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132x−5⋅3x+6=0 13^{2 x}-5 \cdot 3^{x}+6=0 132x−5⋅3x+6=0

$13^{2 x}-5 \cdot 3^{x}+6=0$