Answer :

The student is asking about solving an equation involving logarithms, exponential functions, and trigonometry. Let's break it down and solve it step-by-step.

Given:

[tex]\log_5 3 = t[/tex]

and

[tex]5^{2t-4} = 125[/tex]

We need to find the value of [tex]t[/tex].

Step 1: Solve for [tex]t[/tex] in the first equation.

The first equation is [tex]\log_5 3 = t[/tex]. This can be interpreted as:

[tex]5^t = 3[/tex]

So, [tex]t[/tex] is the power to which 5 must be raised to equal 3.

Step 2: Solve the second equation.

The second equation is [tex]5^{2t-4} = 125[/tex].

We know that 125 is a power of 5. Specifically,

[tex]125 = 5^3[/tex]

Substituting this, the equation becomes:

[tex]5^{2t-4} = 5^3[/tex]

Since the bases are the same, we can equate the exponents:

[tex]2t - 4 = 3[/tex]

Step 3: Solve for [tex]t[/tex].

Add 4 to both sides:

[tex]2t = 7[/tex]

Divide by 2 to solve for [tex]t[/tex]:

[tex]t = \frac{7}{2}[/tex]

Thus, the value of [tex]t[/tex] that satisfies both equations is [tex]\frac{7}{2}[/tex].