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We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mi>s</mi></msup><mi>u</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>ϵ</mi><mi>x</mi><mo>)</mo></mrow><mi>u</mi><mo>=</mo><mi>λ</mi><mi>u</mi><mo>+</mo><mi>u</mi><mo form="prefix">log</mo><msup><mi>u</mi><mn>2</mn></msup><mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace><mrow><mi>in</mi></mrow><mspace width="3.33333pt"></mspace><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>,</mo></mrow></semantics></math></inline-formula> under the mass constraint <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∫</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></msub><msup><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mn>2</mn></msup><mi>d</mi><mi>x</mi><mo>=</mo><mi>a</mi><mo>.</mo></mrow></semantics></math></inline-formula> Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> is an unknown parameter, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mo>−</mo><mo>Δ</mo><mo>)</mo></mrow><mi>s</mi></msup></semantics></math></inline-formula> is the fractional Laplacian and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. We introduce a function space where the energy functional associated with the problem is of class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">C</mi><mn>1</mn></msup></semantics></math></inline-formula>. Then, under some assumptions on the potential <i>V</i> and using the Lusternik–Schnirelmann category, we show that the number of normalized solutions depends on the topology of the set for which the potential <i>V</i> reaches its minimum.