What is 2ln(t)+5ln(z)-ln(t) ?

The solution to 2ln(t)+5ln(z)-ln(t) is ln(tz^5)

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simplify 2ln(t)+5ln(z)-ln(t)

simplify

2 ln (t) + 5 ln (z) - ln (t)

ln (t z^{5})

Solution steps

2 ln (t) + 5 ln (z) - ln (t)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

2 ln (t) = ln (t^{2})

= ln (t^{2}) + 5 ln (z) - ln (t)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

5 ln (z) = ln (z^{5})

= ln (t^{2}) + ln (z^{5}) - ln (t)

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (t^{2}) + ln (z^{5}) = ln (t^{2} z^{5})

= ln (t^{2} z^{5}) - ln (t)

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (t^{2} z^{5}) - ln (t) = ln (\frac{t ^{2} z ^{5}}{t})

= ln (\frac{t ^{2} z ^{5}}{t})

Cancel the common factor:

t

= ln (t z^{5})

lo g_{10} (\frac{85347 x ^{15274}}{125386})

- lo g_{10} (1.74 \times 1 0^{- 5})

lo g_{9} (6 x^{3} y^{5} z)

lo g_{5} (\frac{x ^{2}}{y ^{2} z ^{3}})

lo g_{10} (\frac{7 x}{y ^{2}})