If two equivalent logarithmic expressions have the same base, then the arguments must be equal.
x ≈ 10.61
Practice makes perfect
We want to solve an equation involving more than one logarithm. To do so, we will use the Change of Base Formula to evaluate the given expression.
log_b m = log_c m/log_c b
Numbers m, b, and c are positive numbers, with b≠ 1 and c≠ 1. We will apply this formula with c=9 to simplify one of the expressions.
Next, we will use the fact that if two equivalent logarithmic expressions have the same base, then the arguments must be equal.
log_b x=log_b y ⇔ x= y
Let's apply the above property to our equation.
We will use the Quadratic Formula to solve the given quadratic equation.
ax^2+ bx+ c=0
⇕ x=- b± sqrt(b^2-4 a c)/2 a
We first need to identify the values of a, b, and c.
x^2-14x+36=0
⇕
1x^2+( -14)x+ 36=0
We see that a= 1, b= -14, and c= 36. Let's substitute these values into the Quadratic Formula.
Using the Quadratic Formula, we found that the solutions of the given equation are x=7± sqrt(13). Therefore, the exact solutions are x_1=7+ sqrt(13) and x_2=7-sqrt(13). Let's use a calculator to approximate these values.
x_1 & = 7+ sqrt(13) ≈ 10.61
x_2 & = 7-sqrt(13) ≈ 3.39
To check our answer, we will substitute both 10.61 and 3.39 for x in the given equation one at a time.
Since substituting 10.61 for x in the given equation produces a true statement, x=10.61 is the solution to our equation. Let's now check for the x=3.39.
