Answer:
x ≈ 6.97464
Step-by-step explanation:
To solve the equation [tex]\sf 5^{x-3} = 600[/tex], we can use the logarithmic property that states:
[tex]\sf a^{\log_a(b)} = b[/tex]
In this equation, [tex]\sf a = 5[/tex], [tex]\sf b = 600[/tex], and [tex]\sf x - 3[/tex] is the exponent.
So, we can rewrite the equation as:
[tex]\sf 5^{x-3} = 600[/tex]
[tex]\sf x - 3 = \log_5(600)[/tex]
Now, calculate [tex]\sf \log_5(600)[/tex]:
[tex]\sf x - 3 = \log_5(600) [/tex]
[tex]\sf x - 3 = \dfrac{\log_{10}(600)}{\log_{10}(5)}[/tex]
Using a calculator to find the value of [tex]\sf \log_{10}(600)[/tex] and [tex]\sf \log_{10}(5)[/tex]:
[tex]\sf \log_{10}(600) \approx 2.77815125 [/tex]
[tex]\sf \log_{10}(5) \approx 0.6989700043 [/tex]
[tex]\sf x - 3 = \dfrac{2.77815125}{0.6989700043}[/tex]
[tex]\sf x - 3 \approx 3.974635869[/tex]
Add 3 to both sides:
[tex]\sf x \approx 3 + 3.974635869[/tex]
[tex]\sf x \approx 6.974635869[/tex]
[tex]\sf x \approx 6.97464 \textsf{(in 5 d.p.)}[/tex]
Therefore, x ≈ 6.97464.
