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Changing the subject of an equation involving logarithms often requires the use of the equivalence

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\$\\log_ba=c \\Longleftrightarrow a=b^c\\text{.}\$

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Rearrange the equation to find $x$.

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$\\log_\\var{f}(x)=\\var{f1}$

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$x=$ [[0]]

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Make $x$ the subject of the following equation.

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$\\log_\\var{g1}(x)=y+\\var{g2}$

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$x=$ [[0]]

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Make $x$ the subject of the equation, leaving your answer in the form $a^{\\frac{1}{b}}$.

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$\\log_x(y+\\var{h1})=\\var{h2}$

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$x=$ [[0]]

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$\\log_a(a^x)$

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$a^{\\log_a(x)}$

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$e^{\\ln(x)}$

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$\\log_{10}(x)$

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$\\log_e(x)$

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$\\ln(e^x)$

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Which of the following expressions are equivalent to $x$?

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Rearrange some expressions involving logarithms by applying the relation $\\log_b(a) = c \\iff a = b^c$.

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#### a)

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i)

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We can rearrange logarithms using indices.

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\$\\log_ba=c \\Longleftrightarrow a=b^c\$

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Using this equivalence we can rewrite $\\log_\\var{f}x=\\var{f1}$.

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\\\begin{align} x&= \\var{f}^\\var{f1} \\\\ &=\\var{f^f1} \\end{align}\

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#### b)

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i)

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We can use the equivalence to rewrite our equation.

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\$\\log_ba=c \\Longleftrightarrow a=b^c\$

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We can write out our values to makes it easier.

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\\\begin{align} a&=x \\\\ b&=\\var{g1}\\\\ c&=y+\\var{g2} \\end{align}\

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Then we can write out our equation in the required form.

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\$x=\\var{g1}^{y+\\var{g2}}\$

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#### c)

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We can use the same equivalence as in part b)

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\$\\log_ba=c \\Longleftrightarrow a=b^c\$

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We have

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\\begin{align}
a&=y+\\var{h1} \\\\
b&=x\\\\
c&=\\var{h2}\\text{.} \\\\ \\\\
\\log_{x}(y+\\var{h1}) &= \\var{h2} \\\\
\\implies y+\\var{h1} &= x^{\\var{h2}} \\\\
x &= (y+\\var{h1})^{\\frac{1}{\\var{h2}}}
\\end{align}

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#### d)

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The two in this list that don't equal $x$ are $\\log_e(x)$ and $\\log_{10}(x)$.

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\\\begin{align} \\log_e(x)&=\\ln(x)\\\\ \\log_{10}(x)&=\\log(x)\\text{.} \\end{align}\

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