Answer:
(a) x ≈ -0.93
(d) x ≈ 1.06
Step-by-step explanation:
You want the approximate solutions to log₅(x+5) = x².
We find solving an equation of this nature graphically to be quick and easy. First, we rewrite the equation as ...
log₅(x+5) - x² = 0
Then we graph the left-side expression and let the graphing calculator show us the zeros.
x ≈ -0.93, 1.06
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Additional comment
We can evaluate the above expression for the different answer choices and choose the x-values that make the value of it near zero. The second attachment shows that -0.93 and 1.06 give values with magnitude less than 0.01.
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We calculate that the approximate solutions of the equation [tex]log5(x + 5) = x^2[/tex] are x ≈ -0.93 and x ≈ 1.06.
To find the approximate solutions of the equation, we need to analyze the behavior of the given equation. The equation involves a logarithm and a quadratic term.
First, we can observe that the logarithm has a base of 5 and the argument is x + 5. This means that the value inside the logarithm should be positive for the equation to be defined. Hence, x + 5 > 0, which implies x > -5.
Next, we notice that the right-hand side of the equation is [tex]x^2[/tex], a quadratic term. Quadratic equations typically have two solutions, so we expect to find two approximate solutions.
To determine these solutions, we can use numerical methods or approximations. By analyzing the equation further, we find that the two approximate solutions are x ≈ -0.93 and x ≈ 1.06.
These values satisfy the given equation log5(x + 5) = [tex]x^2[/tex], and they fall within the valid range of x > -5. Therefore, the approximate solutions of the equation are x ≈ -0.93 and x ≈ 1.06.
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